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(0) Comments | Posted February 26, 2015 | 10:45 AM
Fembots, robots and aliens arrive on planet Earth, and they are here to stay. Get used to it! They are fierce and powerful.
Do visit them at the Mia Fonssagrives Solow show at Kasher|Potamkin Gallery at 515 W. 26th Street in New York...
(0) Comments | Posted February 20, 2015 | 5:32 PM
Shades of Grey are also in the air in Paris. Not in beds but in stores, as the shades of grey are the number one happening fashion color for pre spring for women and for men.
The new way to wear grey is to mix it with all the various...
(2) Comments | Posted September 22, 2014 | 11:05 PM
Born in Philadelphia LeGrand Leseur went to Anna Maria college in Massachusetts to study music. He auditioned as a death metal musician, having written his song for his application. LeGrand, much like many ambitious creative artists found studying music in college totally un-inspirational....
(0) Comments | Posted August 17, 2014 | 3:21 PM
Hermes was right, orange is the number one, the chicest color for fashion and accessories. It is the signature color of their shopping bag . Lucky Siri has her own Hermes bag!
This summer of 2014 Hermes Orange is the color for...
(1) Comments | Posted August 7, 2014 | 2:21 PM
The picture perfect romantic getaway town of Claryville, New York has opened its Art Center this weekend with its first show, the work of acclaimed Russian-American artist, Alexander Kaletski titled "Forty Years in America." The show lasts thru the long snowy winter.
(1) Comments | Posted July 23, 2014 | 2:50 PM
Fifty years ago this week, I was twenty and I showed the mini dress in Paris Couture with my best friend Mia.
How and why did two American girls have the COURAGE to cross the Atlantic in May 1964 on an ocean-liner and think France was waiting for their...
(2) Comments | Posted July 2, 2014 | 1:49 PM
Diversity is the name of the fashion game for 2014. I counted 12 entirely different body shapes on women in New York's Penn Station this morning as I went to work on a hot July morning on 7th Ave. The ladies were moving fast going to work, but I managed...
(1) Comments | Posted June 13, 2014 | 5:22 PM
It's 2007 and YSL, Pierre Berge, Sonia Rykiel, Emmanuelle Khahn and myself, all old and wrinkled and not so fabulous anymore are all seated par hazard on the same banquette in Paris. We were all five, by accident, eating at the Brasserie Lipp. It's impossible, but true. I thought I'd...
(0) Comments | Posted June 2, 2014 | 3:34 PM
$Art has taken over the art world; what's a real artist to do?
Contemporary Art is now $Art, and we fine Artists and fine art lovers don't have to take it anymore!
Don't call it contemporary art, call it contemporary non-art, or $Art, and I will say, "Hoorah!"
(0) Comments | Posted May 22, 2014 | 1:18 AM
Where to find your second husband, and it's not online.
Finding husband number two, three or four is very different from finding husband number one as your first husband is made up of innocent dreams and idealistic expectations. Husbands after number one are a whole other ball game.
I have...
(2) Comments | Posted May 8, 2014 | 2:00 PM
I am celebrating 50 years in fashion in 2014 and am today the longest lasting female designer in Paris and America. HOW? How did I last? By what Miracle am I still sewing Vicky Tiel in the back of dresses.
(1) Comments | Posted March 19, 2014 | 1:00 AM
There are two types of marriages, the piggy back marriage or the marriage with two people side by side but each on separate paths. Side by siders can't piggy back, their paths can separate, often for very wide distances and then suddenly they flow back together. I know I'm one...
(0) Comments | Posted January 8, 2014 | 11:45 AM
Children are changing the world. I call this new generation the selfie generation.
My generation, the wonderful sixties revolutionary generation, has returned in spirit today with a new name -- the selfie generation. The selfie has the power, the ambition and the tools to radically change the world....
(0) Comments | Posted December 12, 2013 | 4:45 PM
A magical miracle just happened to me when a turquoise blue ostrich feather fell from heaven and landed on my upstate New York mountaintop cabin doorstep.
I had been up in the log cabin alone for a week designing dresses and resting for my next HSN TV appearance. I...
(1) Comments | Posted December 3, 2013 | 6:31 PM
Beauty in art is in the eye of the beholder and today the beholder loves ugly. I have spent a life of loving art and am sad to comment that today contemporary art (more than any other period of human time) loves ugly.
Van Gogh and Rembrandt had very sad...
(0) Comments | Posted November 5, 2013 | 12:56 PM
Paris shop windows have been invaded by sixties style little red coats. They brighten up the grey tweeds and Mondrian color block stripe patterns of the big coats of last year.
They are worn mini sixties style, four inches above the knee, with classic set in sleeves and...
(6) Comments | Posted October 30, 2013 | 4:11 PM
70 is the new 45. I should know; I just celebrated my 70th birthday with a gift to myself -- a week in Paris all alone, just to eat, look at the fashion in the windows and art in the galleries and to photograph it all for memory's sake.
(0) Comments | Posted October 15, 2013 | 2:15 PM
My recent visit to the Irving Penn exhibit "Assignment " at Pace McGill gallery in NY was a real shock. I expected to see Penn's famous Vogue photos of his wife Lisa, and his favorite model Jean Patchett and his colored still life's, but I certainly did not expect to...
(0) Comments | Posted October 3, 2013 | 3:44 PM
An avocado green can electric opener sold on the radio in Clearwater Florida in 1977 was the birth of at-home shopping. I was dumbfounded when I learned this fact at my first class of "How to Sell on TV" taught in St Petersburg, Florida by Bob Circosta at his corporate...
(1) Comments | Posted August 22, 2013 | 12:25 PM
The sexual revolution started in the late fifties in Paris with two movies and two big movie stars: Brigitte Bardot in And God Created a Woman in 1956 and Jean Paul Belmondo in Breathless in 1959.
Brigitte had real sex (to completion) on camera with her co star and...


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- Shopping Bag ( 0 items )
Customer Reviews for
Puppy Love 3: Reawakening
Page 1 of 1
Posted July 17, 2014
This was a bit of a love fest, annoyingly so at times, but I am
This was a bit of a love fest, annoyingly so at times, but I am glad I read it.Was this review helpful? Yes NoThank you for your feedback. Report this reviewThank you, this review has been flagged.
Don't get me wrong there were some smoking hot scenes but they didn't have the edge the first two books had.
Nothing about this was challenging and I missed that. It made this book almost vanilla.
But it was a sweet end to the trilogy though even if my teeth do ache a bit.
Page 1 of 1

subset: commoncrawl


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The amanda pearl amanda pearl vintage crystal with melon grosgrain bracelet is sold out or discontinued. We found 182 related products.
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Faux Croc Scrapbook Album 12"X12"-Purple
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Faux Croc Scrapbook Album 12"X12"-Black
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The Amanda features added depth, a removable commuter comfort system clear gel insole with cow leather topcover, a thermo rubber outsole with flexible commuter comfort system, a dual density heel with EVA for shock absorption, a tempered steel shank, firm heel counter and accommodates prescribed orthotics.
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The Amanda sandal features a 6 stiletto heel, 2 platform with a two layer lace ankle cuff and back zipper.
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Amanda Lambright loves Wyman Brubaker, and after four years as a single mother, she is grateful for his support and for this new chance at happiness as his wife. She's confident that their children will get along just fine. But once Amanda's clan moves into Wyman's home, the tight quarters and Wyman's reluctance to make changes to accommodate Amanda cause friction. The older kids are squabbling. The little ones are frequently in tears. Tiny Alice Ann isn't speaking at all. Amanda and Wyman can't find any privacy. And Amanda wonders if she'll ever have a chance to pursue the pottery making that means so much to her.
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Color: Purple Faux crocodile print Material: Album board Album board prevents warping and provides durability to last for years Acid and lignin-free PVC-free, archival quality Album includes 100 pages Capacity: 200 4x6-inch photos Dimensions: 8.5 inches high x 8.5 inches wide x 2 inches deep Imported
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aries383 Store This auction is for 1 New Amanda Blu 8-3/4 x 8-3/4 x 2-1/4 in Fabric Photo Album Name of album : Circus Faces These high quality photo albums feature unique stitched details; a durable fabric covering; and are acid and lignin free; PVC free and archival quality. Each book bound album has 100 pages; holds up to 200 4x6 photos with a journaling spot for each photo. Satisfaction Guaranteed !!! Please ck my other auctions for more great deals !!! I will gladly combine shipping on any
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This versatile slide sandal delivers comfort and style. Features metal rhinestone ornamentaion and a 1.75 inch wedge heel. The Amanda will feel as good at the end of the day as it did at the start.
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Amanda Rich Long Sleeve Long Gown (107). This Amanda Rich gown is a more feminine version of the Lucie Ann Classic short sleeve. This Amanda Rich nightie is delicately pleated along the scoop neckline with flower accent and lace hem. Long, full sleeves have an elastic trim with lace bell cuff. Bias cut. Cut with a wide sweep which won't tangle while sleeping. Made in anti-static tricot nylon. Made in USA.
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Plow & Hearth
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DCG Stores
DCG Stores
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Sears Marketplace
Sears Marketplace
d1anadiane Store Princess Amanda Felicity Frog Simulated Pearl Ring Sz 6 BRAND NEW! COMES IN ORIGINAL BOX WITH PAPERS Our charming amphibian comes with a sweet story that starts in a Japanese lotus pond and travels all the way to your jewelry box.Princess Amanda ends the tale with playful whimsy and modest size that's perfect for everyday wear.Good thing, too, because you'll never want to take it off. Design Information Dimensional, frog-on-lily pad design Pavé-set, round green crystals encr
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Listed is a New Amanda Blu Fabric Embroidered Boy's Photo Album and an Assortment of Scrapbooking Goodies to enhance your memory making fun! It's such an assortment of very cute accessories to help you begin to create your special album! Amanda Blu Albums are not only adorable, but very well made.This lot includes the following:(These items are in the same order as the listed photos).An Amanda Blu 12 x 14 fabric embroidered album. This album is new and unused however I did remove it from the
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AMDR1019 Features Crystal ivory Cultured Pearls and crystal pink coral Cultured Pearls adorn this thicker gold-plated brass chain Each piece is handmade with great attention to detail Made in the USA Color: Pink Necklace Style: Strands Material: Brass Length: Over 20'' Color: White Color: Pink Stone Type: Coral/Crystal Gender: Women's Lifestage: Adults Generic Dimensions Chain length: 36'', no clasp Item weight: 0.5 lb Dimensions Overall Product Weight: 0.5 Pounds
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d1anadiane Store Princess Amanda Felicity Frog Simulated Pearl Ring Sz 7 BRAND NEW! COMES IN ORIGINAL BOX WITH PAPERS Our charming amphibian comes with a sweet story that starts in a Japanese lotus pond and travels all the way to your jewelry box.Princess Amanda ends the tale with playful whimsy and modest size that's perfect for everyday wear.Good thing, too, because you'll never want to take it off. Design Information Dimensional, frog-on-lily pad design Pavé-set, round green crystals encr
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cashaw47 Store Amanda Lawford Elegant Oriental Rooster Hen handpainted Needlepoint Canvas Description:Welcome to Needlepoint By Wildflowers ! My goal is to offer you the finest hand painted needlepoint canvases from the most talented artists, stitching supplies, accessories and Professional Services. Offered for sale is an Elegant *NEW* Design from Amanda Lawford, titled Oriental Rooster and Hen and is hand painted in elegant shades with a soft tan background ~ hand painte
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AMDR1016 Features One crystal lapis Cultured Pearl hangs beneath a cubic zirconia earring post Each piece is handmade with great attention to detail Made in the USA Timeless Beauty collection Style: Drop Earrings Stone: Pearl Color: Blue Closure Type: Post and Finding Birthstone: June Material: Yellow Gold Gemstone: Pearl Gender: Women's Earring Set: Yes Lifestage: Adults Generic Dimensions Height: 19.05mm
$49 Go to


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Bleeding Gums Do I Need To Get Tested For HIV
Here's a taste of what TheBody.com has to offer on this topic:
Ask the Experts
Gums bleed and I performed anilingus on someone that is HIV+Hi,I believe I asked a question close to this before, but I didn't make clear that my gums bleed often and are very sensitive. i performed aniligus on a partner that we found out later has HIV.. I only used my tongue and we we didn't do anything...
Response from Robert J. Frascino, M.D.
The Robert James Frascino AIDS Foundation...relevant point here. You clearly already are worried. The bottom line is always: if you feel you've placed yourself at risk for HIV, get a single HIV-antibody test at the three-month mark. Your HIV-acquisition risk remains very low.Good luck.Dr.... Read more »
Bleeding Gums and Unprotected anal sexHi Dr Bob,As the title suggests I do have couple of issues bothering me at the moment. I was in... Read more »
OMG, My Dr says I need PEP, please help?Hi Dr Bob I had gone through your forum before seeing my doctor and now I don't know what to... Read more »
Safer Sex Guidelines...more difficult for blood or sexual fluid to get into your body. Be aware of your body and your partner's. Cuts, sores, or bleeding gums increase the risk of spreading HIV. Rough physical activity also increases the risk. Even small injuries... Read more »
Hot Topics at The Body's "Ask the Experts" Forums, October 19, 2010...I got bitten by my assistant who's HIV positive. The skin on my finger tore when I pulled it out of his teeth, but he had no bleeding gums or sores in his mouth. Should I be worried about becoming HIV positive? STRANGE BUT TRUE ... Read more »
Women, Anemia and HIV/AIDS... Table of Contents What Is Anemia? Anemia and HIV What Causes Anemia? Diagnosis and Treatment Taking Care of Yourself What Is Anemia? Anemia is a medical problem Read more »
What Is AIDS?...infected with HIV through oral sex or in rare cases through deep kissing, especially if you have open sores in your mouth or bleeding gums. For more information, see the following Fact Sheets: 150: Stopping the Spread of HIV 151: Safer Sex... Read more »
Connect With Others
anyone help pls!!
Posted by Anonymous, 3 Replies...some of the sites out there that says insertive oral sex does carry risk if the person performing it has bad oral health such a bleeding gums. So my question is do i really need to get a HIV test done? i had a glance at the condom after the... Read more »
Quick Question...
Posted by Anonymous, 2 Replies...HIV? I would think that it would put the person that I am doing it to at a greater risk because they are recieving some of my blood from my gums...what I want to know is if I am at risk too because of the bleeding? Read more »
Posted by MEguy, 2 Replies...apart from oral. I know I need to get tested, but have to say am very nervous about going. I guess my question is: Are bleeding gums a symptom of HIV or any other STD?? Thanks Read more »
Oral sex risk
Posted by Anonymous, 1 Reply...Hey guys im female and performed oral sex on a guy a few weeks ago. I read that the risk is increased if one has bleeding gums. well i recently had a root canal done and the gums around the tooth have been sore and a little dry blood on the gums... Read more »
Resource Centers
Guides for People With HIV


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- 25Promote
Art Print
Decoby Lyle Hatch
Collect your choice of gallery quality Giclée, or fine art prints custom trimmed by hand in a variety of sizes with a white border for framing.
Exceptional work Lyle
Hey John - glad you like it!
Nice 1 Lyle!!!!! What prog do you use???
Thanks Lyle....did try it, but had no joy!!!!! :-( Must have another go. I use Incendia as I've got my head around it...A BIT!!!!! Shall revisit and see what happens??????? Thanks Lyle!!
All the fractal and 3D programs I've ever tried seem to have different learning curves - all pretty steep. We just have to keep trying and experimenting, I guess. Good luck, Brian!
Thanks Lyle....I'll need it lol :-)
Cool abstract!
Very nice!!
See More Comments


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If you haven't been following the conversation around Occupy Wall
Street, it's perhaps best summarized in terms of the Tumblrs.
First there were the 99 percent, who have been demonstrating in
New York and elsewhere for weeks.
blogging about blogging can be the ultimate navel-gazing, but hear me out; I
mean to intrigue.


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Stir-fry ham, onion, and green pepper in butter in a heavy 9" skillet over moderate heat 2-3 minutes until onion is limp.
Mix eggs, water, salt, and pepper, pour into skillet and fry like a pancake until firm and lightly browned underneath; turn and brown flip side.
Halve and sandwich between toast, folding as needed to fit.


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Curtis Flannick from Menomonee Falls will compete in the NFL PUNT, PASS & KICK National Finals on Saturday in Denver during the weekend of the AFC Divisional Playoff Game. Flannick was one of 40 young athletes chosen from across the country to participate in the competition.
An on-field award presentation will air after the third quarter of the game, which will be televised Sunday on CBS. Flannick will represent the Packers.
Finalists will compete separately in five age divisions at Valor Christian High School in Highlands Ranch, Colo., with the top scorer in each group crowned national champion. All participants launch two punts, two passes and two kicks with scores based on distance and accuracy. The best score from each activity is tabulated to determine the athlete's total.
The athletes advanced to the national finals through local, sectional and team championship competitions held throughout the NFL regular season. The top four scorers in each age group from across the country qualified as national finalists.
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- Falls-based Bradley Corporation recieves nod for wellness efforts


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Unfortunately MagLite Bulb Protector AA Mini Mag is no longer carried by OpticsPlanet.
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Well met at Audiko, where it's possible to search for ringtones for free for your iPhone. Today we have a ringtone of "I'll Be Missing You" by Puff Daddy as well as 422 other ringtones by Puff Daddy. "I'll Be Missing You" is a top-rated song from the "No Way Out" album that was made by Puff Daddy at their recording studio together with other 9 tracks. You can listen to all 104 Puff Daddy – "I'll Be Missing You" ringtone variations available here and download it for free right here. You can also select other 2 albums produced by Puff Daddy for all time. "I'll Be Missing You" is referred to Hip-Hop, and Audiko offer a huge amount of Hip-Hop collaborations absolutely for free. And finally, we have collected lyrics for this track and you can view it here as well. So feel free to browse, listen, download ringtones at Audiko.net. Make your phone sound cool!
Puff Daddy - I'll Be Missing You
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Abbotsholme - so many photos!
We had such a good course at Abbotsholme School this Easter holiday.
Flutewise at Abbotsholme 2014 - what a course!
Our course at Abbotsholme over the Easter holiday was fantastic. I can't believe how hard everyone worked and how much fun we all had.
Our courses aim to:
Sir James Galway Recital and website
If you live within travelling distance of London you
Amazing course! Amazing prizes!Submitted by Liz on 18 Apr 14, 15:25
Abbotsholme 2014 day 3
7.30 am A peaceful night! Seven young people waiting for Cathy to go for a run. Chris and Emma have already been practising for at least half an hour.
Abbotsholme 2014 Day 2
What a fantastic morning we woke up to! Everyone seemed to have slept well all night and I'm delighted I had no problems.
Abbotsholme day 1
We are here! It is wonderful!
Flutewise at Abbotsholme 2014!
I'm off. After 10 days of preparation, everything is packed and ready to go (I hope)! I'm really looking forward to this course, we've got lots of new, young people coming along.
The Sound of Proper Posture
I'd love to go to this but I can't as I'll be away, maybe you can.
The Sound of Proper Posture
Flutewise Club in Arlesey, March 2014
Flutewise Blog – Flutewise Club in Arlesey, March 2014 – Emma Halnan


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Book your room at St Benedict's Byre
Battle bed & breakfast with free self parking
Collect 200 points
Simply enter your Nectar card number when prompted. Your points will be credited 30 days after you return from your trip.*Offers may apply
Situated in a rural location, this bed & breakfast is close to Blackhorse Quarry, Battle Abbey and Battlefield, and Church of St Mary the Virgin. Local attractions also include Buckleys Yesterday's World and Gardens and Battle Museum of Local History.
Property Features
In addition to complimentary wireless Internet access, St Benedict's Byre provides free self parking, a garden, and tour/ticket assistance.
Room Amenities
Guestrooms include complimentary wireless Internet access, coffee/tea makers, and bathrooms with showers.Information missing or incorrect? Tell us!
Hotel Amenities
Hotel Amenities
St Benedict's Byre features a terrace, tour/ticket assistance, and a garden. Public areas are equipped with complimentary high-speed wireless Internet access. Complimentary self parking is available on site.
St Benedict's Byre is a smoke-free property.
- Breakfast available (surcharge)
- Tours/ticket assistance
- Free WiFi
- Garden
- Terrace
- Free self parking
- Smoke-free property
Available in all rooms: Free WiFi
Available in some public areas: Free WiFi
Free self parking
Room Amenities
- Coffee/tea maker
- Private bathroom
- Shower only
- Iron/ironing board (on request)
- Free WiFi
Where to Eat
Full breakfasts are available for a surcharge.
Nearby Things to Do
The recreational activities listed below are available either on site or nearby; fees may apply.
- Golfing nearby
Hotel Policies
Check-in time starts at 2 PM
Check-in time ends at 9 PM
Check-out time is 10 AM
You need to know
Extra-person charges may apply and vary depending on hotel policy.
Government-issued photo identification and a credit card or cash deposit are required at check-in for incidental charges.
Special requests are subject to availability upon check-in and may incur additional charges. Special requests cannot be guaranteed.


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Arthur Henry Adams
A Song of Failure
HERE is my hand to you, brother,
You of the ruck who have failed
I, too, am only another
Fighter who faltered and quailed.
Now with my courage for token
Here to grim Fate I give tithe;
I, too, am beaten and broken,
Lying, the swath of the scythe!
We to the conquerors' seeming
Crouch, an incongruous horde—
Fighters, enmeshed in their dreaming,
Dreamers who girl on the sword,
Weaklings with splendid ambitions,
Heroes who learnt to succumb,
Poets a-swoon in their visions,
Singers with ecstasy dumb.
Failed! So we cast off our burden,
Done with our doubts and our fears:
These we have won for our guerdon—
Pity and tears—women's tears!
You with your conquests unending
Dwell from a woman apart;
Only the humble and bending
Learn the low door to her heart.
We that lie dumb in your scorning
Made you the heroes you are,
Built you a road to the morning,
Taught you to reach for a star:
We have had sight of the glory,
Pointed it clear to the blind;
Yours is the conquerors' story,
Ours is the vision you find.
Here is no dread and no grieving;
Over us hurtles the fray,
Is yours a Heav'n worth achieving,
If it be stormed in a day?
Here is this world we must live in—
Little to lose or to gain;
More is it worth to have striven
Than in the end to attain.
Arthur Henry Adams's Other Poems
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Comments about this poem (A Song of Failure by Arthur Henry Adams )
Gangadharan Nair Pulingat (4/16/2014 10:07:00 AM)
A self inspirational poem for success in real life and a model poem for self improvement and confidence and in this aspect poem is very nice. (Report) Reply
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Feb 28
Ja Ja Ja presents Burning Hearts, Carmen Villain, Lay Low Thursday, February 28Ja Ja Ja presents Burning Hearts, Carmen Villain, Lay Low7.30pm adv £5 BURNING HEARTS Finnish indie stalwarts Jessika Rapo (vocalist for Le Futur Pompiste) and Henry Ojala (ex-drummer for Cats on Fire) teamed up to form the synth pop duo Burning Hearts in 2006. Burning Hearts' debut full-length 'Aboa Sleeping' received a warm welcome on release in 2009, making it to the top albums of several music blogs that year. Stream the otherworldy sounds of 'Into The Wilderness' as featured on Ja Ja Ja's November mixtape: https://soundcloud.com/jajajanordic/burning-hearts-into-the Burning Hearts are currently running a Christmas Calendar Tumblr, with this cover of legendary Finnish band Leevi and the Leavings'1982 song 'En tahdo sinua enää' (translated as 'I Don't Want You Any More') complete with a karaoke style video being behind the first window: http://youtu.be/xYjlh0L6wmE Watch the atmospheric glittery-green video for 'Modern Times': www.youtube.com/watch?v=BetluZszbHM CARMEN VILLAIN 'Most of my songs are about escaping something��" escaping this weird vacuum, an unsatisfying world' Carmen Villain Watch the as-yet unreleased 'Made A Shell', taken from a blistering performance filmed at Parkteateret: www.youtube.com/watch?v=Q6ojHUCL5u4 LAY LOW Lay Low is the alter-ego of Icelandic singer/songwriter Lovísa Elísabet Sigrúnardóttir. Watch the charming video for recent single 'The Backbone': www.youtube.com/watch?v=SA5YUI3k43k Watch this haunting animated clip for 'Helganga', an Icelandic poem by Þórhildur Sveinsdóttir: www.youtube.com/watch?v=3saDQk-3AV4
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Donald Heiman
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Professor Heiman's research group focuses on several challenging and technologically important areas: (1) topological insulators; (2) nanomagnetism; and (3) novel spintronic materials. The group's primary experimental tools are: molecular beam epitaxy (MBE) crystal growth; SQUID magnetrometry; electrical and optical measurements in high magnetic fields and at low temperatures. Research is in collaboration with MIT and Leeds University (UK), and is supported by the NSF, DOE and ONR.
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text:
Fermat number
In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form
where n is a nonnegative integer. The first few Fermat numbers are:
If 2k + 1 is prime, and k > 0, it can be shown that k must be a power of two. (If k = ab where 1 ≤ a, b ≤ k and b is odd, then 2k + 1 = (2a)b + 1 ≡ (−1)b + 1 = 0 (mod 2a + 1). See below for a complete proof.) In other words, every prime of the form 2k + 1 (other than 2 = 20 + 1 ) is a Fermat number, and such primes are called Fermat primes. The only known Fermat primes are F0, F1, F2, F3, and F4 (sequence A019434 in OEIS).
- 1 Basic properties
- 2 Primality of Fermat numbers
- 3 Factorization of Fermat numbers
- 4 Pseudoprimes and Fermat numbers
- 5 Selfridge's Conjecture
- 6 Other theorems about Fermat numbers
- 7 Relationship to constructible polygons
- 8 Applications of Fermat numbers
- 9 Other interesting facts
- 10 Generalized Fermat numbers
- 11 See also
- 12 Notes
- 13 References
- 14 External links
Basic properties[edit]
The Fermat numbers satisfy the following recurrence relations:
for n ≥ 1,
for n ≥ 2. Each of these relations can be proved by mathematical induction. From the last equation, we can deduce Goldbach's theorem (named after Christian Goldbach): no two Fermat numbers share a common integer factor greater than 1. To see this, suppose that 0 ≤ i < j and Fi and Fj have a common factor a > 1. Then a divides both
and Fj; hence a divides their difference, 2. Since a > 1, this forces a = 2. This is a contradiction, because each Fermat number is clearly odd. As a corollary, we obtain another proof of the infinitude of the prime numbers: for each Fn, choose a prime factor pn; then the sequence {pn} is an infinite sequence of distinct primes.
Further properties:
- The number of digits D(n,b) of Fn expressed in the base b is
- (See floor function).
- No Fermat number can be expressed as the sum of two primes, with the exception of F1 = 2 + 3.
- No Fermat prime can be expressed as the difference of two pth powers, where p is an odd prime.
- With the exception of F0 and F1, the last digit of a Fermat number is 7.
- The sum of the reciprocals of all the Fermat numbers (sequence A051158 in OEIS) is irrational. (Solomon W. Golomb, 1963)
Primality of Fermat numbers[edit]
Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured (but admitted he could not prove) that all Fermat numbers are prime. Indeed, the first five Fermat numbers F0,...,F4 are easily shown to be prime. However, this conjecture was refuted by Leonhard Euler in 1732 when he showed that
Euler proved that every factor of Fn must have the form k2n+1 + 1 (later improved to k2n+2 + 1 by Lucas).
The fact that 641 is a factor of F5 can be easily deduced from the equalities 641 = 27×5+1 and 641 = 24 + 54. It follows from the first equality that 27×5 ≡ −1 (mod 641) and therefore (raising to the fourth power) that 228×54 ≡ 1 (mod 641). On the other hand, the second equality implies that 54 ≡ −24 (mod 641). These congruences imply that −232 ≡ 1 (mod 641).
It is widely believed that Fermat was aware of the form of the factors later proved by Euler, so it seems curious why he failed to follow through on the straightforward calculation to find the factor.[1] One common explanation is that Fermat made a computational mistake and was so convinced of the correctness of his claim that he failed to double-check his work.
There are no other known Fermat primes Fn with n > 4. However, little is known about Fermat numbers with large n.[2] In fact, each of the following is an open problem:
- Is Fn composite for all n > 4?
- Are there infinitely many Fermat primes? (Eisenstein 1844)[3]
- Are there infinitely many composite Fermat numbers?
- Does a Fermat number exist that is not square-free?
As of 2014[update], it is known that Fn is composite for 5 ≤ n ≤ 32, although amongst these, complete factorizations of Fn are known only for 0 ≤ n ≤ 11, and there are no known factors for n = 20 and n = 24.[4] The largest Fermat number known to be composite is F3329780, and its prime factor 193 × 23329782 + 1, a megaprime, was discovered by the PrimeGrid collaboration in July 2014.[4][5]
Heuristic arguments for density[edit]
The following heuristic argument suggests there are only finitely many Fermat primes: according to the prime number theorem, the "probability" that a number n is prime is at most A/ln(n), where A is a fixed constant. Therefore, the total expected number of Fermat primes is at most
It should be stressed that this argument is in no way a rigorous proof. For one thing, the argument assumes that Fermat numbers behave "randomly", yet we have already seen that the factors of Fermat numbers have special properties. If (more sophisticatedly) we regard the conditional probability that n is prime, given that we know all its prime factors exceed B, as at most Aln(B)/ln(n), then using Euler's theorem that the least prime factor of Fn exceeds 2n + 1, we would find instead
Although such arguments engender the belief that there are only finitely many Fermat primes, one can also produce arguments for the opposite conclusion. Suppose we regard the conditional probability that n is prime, given that we know all its prime factors are 1 modulo M, as at least CM/ln(n). Then using Euler's result that M = 2n + 1 we would find that the expected total number of Fermat primes was at least
and indeed this argument predicts that an asymptotically constant fraction of Fermat numbers are prime.
Equivalent conditions of primality[edit]
There are a number of conditions that are equivalent to the primality of Fn.
- Proth's theorem (1878)—Let N = k2m + 1 with odd k < 2m. If there is an integer a such that
- then N is prime. Conversely, if the above congruence does not hold, and in addition
- (See Jacobi symbol)
- then N is composite. If N = Fn > 3, then the above Jacobi symbol is always equal to −1 for a = 3, and this special case of Proth's theorem is known as Pépin's test. Although Pépin's test and Proth's theorem have been implemented on computers to prove the compositeness of some Fermat numbers, neither test gives a specific nontrivial factor. In fact, no specific prime factors are known for n = 20 and 24.
- Let n ≥ 3 be a positive odd integer. Then n is a Fermat prime if and only if for every a co-prime to n, a is a primitive root modulo n if and only if a is a quadratic nonresidue modulo n.
- The Fermat number Fn > 3 is prime if and only if it can be written uniquely as a sum of two nonzero squares, namely
- When not of the form shown above, a proper factor is:
- Example 1: F5 = 622642 + 204492, so a proper factor is
- Example 2: F6 = 40468032562 + 14387937592, so a proper factor is
Factorization of Fermat numbers[edit]
Because of the size of Fermat numbers, it is difficult to factorize or even to check primality. Pépin's test gives a necessary and sufficient condition for primality of Fermat numbers, and can be implemented by modern computers. The elliptic curve method is a fast method for finding small prime divisors of numbers. Distributed computing project Fermatsearch has successfully found some factors of Fermat numbers. Yves Gallot's proth.exe has been used to find factors of large Fermat numbers. Édouard Lucas, improving the above-mentioned result by Euler, proved in 1878 that every factor of Fermat number , with n at least 2, is of the form (see Proth number), where k is a positive integer. By itself, this is almost sufficient to prove the primality of the known Fermat primes.
Factorizations of the first twelve Fermat numbers are:
|3 is prime
|5 is prime
|17 is prime
|257 is prime
|65,537 is the largest known Fermat prime
|641 × 6,700,417
|18,446,744,073,709,551,617 (20 digits)
|274,177 × 67,280,421,310,721 (14 digits)
|340,282,366,920,938,463,463,374,607,431,768,211,457 (39 digits)
|59,649,589,127,497,217 (17 digits) × 5,704,689,200,685,129,054,721 (22 digits)
639,937 (78 digits)
|1,238,926,361,552,897 (16 digits) ×
93,461,639,715,357,977,769,163,558,199,606,896,584,051,237,541,638,188,580,280,321 (62 digits)
649,006,084,097 (155 digits)
|2,424,833 × 7,455,602,825,647,884,208,337,395,736,200,454,918,783,366,342,657 (49 digits) ×
504,705,008,092,818,711,693,940,737 (99 digits)
|179,769,313,486,231,590,772,930,...,304,835,356,329,624,224,137,217 (309 digits)
|45,592,577 × 6,487,031,809 × 4,659,775,785,220,018,543,264,560,743,076,778,192,897 (40 digits) ×
130,439,874,405,488,189,727,484,...,806,217,820,753,127,014,424,577 (252 digits)
|323,170,060,713,110,073,007,148,...,193,555,853,611,059,596,230,657 (617 digits)
|319,489 × 974,849 × 167,988,556,341,760,475,137 (21 digits) × 3,560,841,906,445,833,920,513 (22 digits) ×
173,462,447,179,147,555,430,258,...,491,382,441,723,306,598,834,177 (564 digits)
As of 2014[update], only F0 to F11 have been completely factored.[4] The distributed computing project Fermat Search is searching for new factors of Fermat numbers.[6] The set of all Fermat factors is A050922 (or, sorted, A023394) in OEIS.
The following factors of Fermat numbers were known before 1950 (since the 1950s digital computers have helped find more factors):
|Fermat number
|(fully factored)
|(fully factored)
As of March 2014[update], 315 prime factors of Fermat numbers are known, and new ones are discovered each year.
Pseudoprimes and Fermat numbers[edit]
Like composite numbers of the form 2p − 1, every composite Fermat number is a strong pseudoprime to base 2. This is because all strong pseudoprimes to base 2 are also Fermat pseudoprimes - i.e.
for all Fermat numbers.
It is generally believed that all but the first few Fermat numbers are composite. If proven true, this would mean it is possible to generate infinitely many strong pseudoprimes to base 2 from the Fermat numbers.
In 1964, Rotkiewicz showed that the product of at least two prime or composite Fermat numbers will be a Fermat pseudoprime to base 2.
Selfridge's Conjecture[edit]
John L. Selfridge made an intriguing conjecture. Let g(n) be the number of distinct prime factors of 22n + 1 (sequence A046052 in OEIS). Then g(n) is not monotonic (nondecreasing). If another Fermat prime exists, that would imply the conjecture.[7]
Other theorems about Fermat numbers[edit]
Lemma: If n is a positive integer,
Theorem: If is an odd prime, then is a power of 2.
If is a positive integer but not a power of 2, then where , and r and s are coprime. Without loss of generality let be odd.
By the preceding lemma, for positive integer ,
where means "evenly divides". Substituting , , and and using that is odd,
and thus
Because , it follows that is not prime. Therefore, by contraposition must be a power of 2.
Theorem: A Fermat prime cannot be a Wieferich prime.
Proof: We show if is a Fermat prime (and hence by the above, m is a power of 2), then the congruence does not hold.
It is easy to show . Now write . If the given congruence holds, then , and therefore
Hence ,and therefore . This leads to
, which is impossible since .
A theorem of Édouard Lucas: Any prime divisor p of Fn = is of the form whenever n is greater than one.
Sketch of proof:
Let Gp denote the group of non-zero elements of the integers (mod p) under multiplication, which has order p-1. Notice that 2 (strictly speaking, its image (mod p)) has multiplicative order dividing in Gp (since is the square of which is -1 mod Fn), so that, by Lagrange's theorem, p-1 is divisible by and p has the form for some integer k, as Euler knew. Édouard Lucas went further. Since n is greater than 1, the prime p above is congruent to 1 (mod 8). Hence (as was known to Carl Friedrich Gauss), 2 is a quadratic residue (mod p), that is, there is integer a such that a2 -2 is divisible by p. Then the image of a has order in the group Gp and (using Lagrange's theorem again), p-1 is divisible by and p has the form for some integer s.
In fact, it can be seen directly that 2 is a quadratic residue (mod p), since (mod p). Since an odd power of 2 is a quadratic residue (mod p), so is 2 itself.
Relationship to constructible polygons[edit]
An n-sided regular polygon can be constructed with compass and straightedge if and only if n is the product of a power of 2 and distinct Fermat primes: in other words, if and only if n is of the form n = 2kp1p2…ps, where k is a nonnegative integer and the pi are distinct Fermat primes.
A positive integer n is of the above form if and only if its totient φ(n) is a power of 2.
Applications of Fermat numbers[edit]
Pseudorandom Number Generation[edit]
Fermat primes are particularly useful in generating pseudo-random sequences of numbers in the range 1 … N, where N is a power of 2. The most common method used is to take any seed value between 1 and P − 1, where P is a Fermat prime. Now multiply this by a number A, which is greater than the square root of P and is a primitive root modulo P (i.e., it is not a quadratic residue). Then take the result modulo P. The result is the new value for the RNG.
This is useful in computer science since most data structures have members with 2X possible values. For example, a byte has 256 (28) possible values (0–255). Therefore to fill a byte or bytes with random values a random number generator which produces values 1–256 can be used, the byte taking the output value − 1. Very large Fermat primes are of particular interest in data encryption for this reason. This method produces only pseudorandom values as, after P − 1 repetitions, the sequence repeats. A poorly chosen multiplier can result in the sequence repeating sooner than P − 1.
Other interesting facts[edit]
A Fermat number cannot be a perfect number or part of a pair of amicable numbers. (Luca 2000)
The series of reciprocals of all prime divisors of Fermat numbers is convergent. (Křížek, Luca & Somer 2002)
If nn + 1 is prime, there exists an integer m such that n = 22m. The equation nn + 1 = F(2m+m) holds at that time.[8]
Let the largest prime factor of Fermat number Fn be P(Fn). Then,
Generalized Fermat numbers[edit]
Numbers of the form , where a > 1 are called generalized Fermat numbers. An odd prime p is a generalized Fermat number if and only if p is congruent to 1 (mod 4). (Here we consider only the case n>0, so 3 = is not a counterexample.)
By analogy with the ordinary Fermat numbers, it is common to write generalized Fermat numbers of the form as Fn(a). In this notation, for instance, the number 100,000,001 would be written as F3(10). In the following we shall restrict ourselves to primes of this form, .
If we require n>0, then Landau's fourth problem asks if there are infinitely many generalized Fermat primes Fn(a).
Generalized Fermat primes[edit]
Because of the ease of proving their primality, generalized Fermat primes have become in recent years a hot topic for research within the field of number theory. Many of the largest known primes today are generalized Fermat primes.
Generalized Fermat numbers can be prime only for even a, because if a is odd then every generalized Fermat number will be divisible by 2. By analogy with the heuristic argument for the finite number of primes among the base-2 Fermat numbers, it is to be expected that there will be only finitely many generalized Fermat primes for each even base. The smallest prime number Fn(a) with n > 4 is F5(30), or 3032+1.
The smallest base b such that b2n + 1 is prime are
- 2, 2, 2, 2, 2, 30, 102, 120, 278, 46, 824, 150, 1534, 30406, 67234, 70906, 48594, 62722, 24518, 75898, ... (sequence A056993 in OEIS)
The smallest k such that (2n)k + 1 is prime are
- 1, 1, 1, 0, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 1, 0, 4, 1, ... (The next term is unknown) (sequence A079706 in OEIS)
A more elaborate theory can be used to predict the number of bases for which Fn(a) will be prime for a fixed n. The number of generalized Fermat primes can be roughly expected to halve as n is increased by 1.
Largest known generalized Fermat primes[edit]
The following is a list of the 10 largest known generalized Fermat primes.[9] They are all megaprimes. As of February 2015[update] the whole top-10 was discovered by participants in the PrimeGrid project.
|Prime rank[10]
|Prime number
|Generalized Fermat notation
|Number of digits
|Found date
|475856524288 + 1
|2012 August 8
|356926524288 + 1
|2012 June 20
|341112524288 + 1
|2012 June 15
|75898524288 + 1
|2011 November 19
|773620262144 + 1
|2012 April 19
|676754262144 + 1
|2012 February 12
|525094262144 + 1
|2012 January 18
|361658262144 + 1
|2011 October 29
|145310262144 + 1
|2011 February 8
|40734262144 + 1
|2011 March 8
See also[edit]
- Constructible polygon: which regular polygons are constructible partially depends on Fermat primes.
- Double exponential function
- Lucas' theorem
- Mersenne prime
- Pierpont prime
- Primality test
- Proth's theorem
- Pseudoprime
- Sierpiński number
- Sylvester's sequence
- ^ Křížek, Luca & Somer 2001, p. 38, Remark 4.15
- ^ Chris Caldwell, "Prime Links++: special forms" at The Prime Pages.
- ^ Ribenboim 1996, p. 88.
- ^ a b c Keller, Wilfrid (February 7, 2012), "Prime Factors of Fermat Numbers", ProthSearch.net, retrieved March 5, 2012
- ^ "PrimeGrid's Mega Prime Search – 193*2^3329782+1 (official announcement)". PrimeGrid. Retrieved 7 August 2014.
- ^ FermatSearch.org
- ^ Prime Numbers: A Computational Perspective, Richard Crandall and Carl Pomerance, Second edition, Springer, 2011 Look up Selfridge's Conjecture in the Index.
- ^ Jeppe Stig Nielsen, "S(n) = n^n + 1".
- ^ Caldwell, Chris K. "The Top Twenty: Generalized Fermat". The Prime Pages. Retrieved 6 February 2015.
- ^ Caldwell, Chris K. "The Top Twenty: Generalized Fermat". The Prime Pages. Retrieved 6 February 2015.
- ^ "PrimeGrid's Generalized Fermat Prime Search - 475856^524288+1". Primegrid. Retrieved 21 August 2012.
- ^ "PrimeGrid's Generalized Fermat Prime Search - 356926^524288+1". Primegrid. Retrieved 30 July 2012.
- ^ "PrimeGrid's Generalized Fermat Prime Search - 341112^524288+1". Primegrid. Retrieved 9 July 2012.
- ^ "PrimeGrid's Generalized Fermat Prime Search - 75898^524288+1". Primegrid. Retrieved 9 July 2012.
- ^ "PrimeGrid's Generalized Fermat Prime Search - 773620^262144+1". Primegrid. Retrieved 9 July 2012.
- ^ "PrimeGrid's Generalized Fermat Prime Search - 676754^262144+1". Primegrid. Retrieved 9 July 2012.
- ^ "PrimeGrid's Generalized Fermat Prime Search - 525094^262144+1". Primegrid. Retrieved 9 July 2012.
- ^ "PrimeGrid's Generalized Fermat Prime Search - 361658^262144+1". Primegrid. Retrieved 9 July 2012.
- ^ "PrimeGrid's Generalized Fermat Prime Search - 145310^262144+1". Primegrid. Retrieved 9 July 2012.
- ^ "PrimeGrid's Generalized Fermat Prime Search - 40734^262144+1". Primegrid. Retrieved 9 July 2012.
- Golomb, S. W. (January 1, 1963), "On the sum of the reciprocals of the Fermat numbers and related irrationalities" (PDF), Canadian Journal of Mathematics (Canadian Mathematical Society) 15: 475–478, doi:10.4153/CJM-1963-051-0, ISSN 0008-414X
- Grytczuk, A.; Luca, F. & Wójtowicz, M. (2001), "Another note on the greatest prime factors of Fermat numbers", Southeast Asian Bulletin of Mathematics (Springer-Verlag) 25 (1): 111–115, doi:10.1007/s10012-001-0111-4, ISSN 0129-2021
- Guy, Richard K. (2004), Unsolved Problems in Number Theory, Problem Books in Mathematics 1 (3rd ed.), New York: Springer Verlag, pp. A3, A12, B21, ISBN 0-387-20860-7
- Křížek, Michal; Luca, Florian & Somer, Lawrence (2001), 17 Lectures on Fermat Numbers: From Number Theory to Geometry, CMS books in mathematics 10, New York: Springer, ISBN 0-387-95332-9 - This book contains an extensive list of references.
- Křížek, Michal; Luca, Florian & Somer, Lawrence (2002), "On the convergence of series of reciprocals of primes related to the Fermat numbers" (PDF), Journal of Number Theory (Elsevier) 97 (1): 95–112, doi:10.1006/jnth.2002.2782, ISSN 0022-314X
- Luca, Florian (2000), "The anti-social Fermat number", American Mathematical Monthly (Mathematical Association of America) 107 (2): 171–173, doi:10.2307/2589441, ISSN 0002-9890
- Ribenboim, Paulo (1996), The New Book of Prime Number Records (3rd ed.), New York: Springer, ISBN 0-387-94457-5
- Robinson, Raphael M. (1954), "Mersenne and Fermat Numbers", Proceedings of the American Mathematical Society (American Mathematical Society) 5 (5): 842–846, doi:10.2307/2031878
- Yabuta, M. (2001), "A simple proof of Carmichael's theorem on primitive divisors" (PDF), Fibonacci Quarterly (Fibonacci Association) 39: 439–443, ISSN 0015-0517
External links[edit]
- Chris Caldwell, The Prime Glossary: Fermat number at The Prime Pages.
- Luigi Morelli, History of Fermat Numbers
- John Cosgrave, Unification of Mersenne and Fermat Numbers
- Wilfrid Keller, Prime Factors of Fermat Numbers
- Weisstein, Eric W., "Fermat Number", MathWorld.
- Weisstein, Eric W., "Fermat Prime", MathWorld.
- Weisstein, Eric W., "Fermat Pseudoprime", MathWorld.
- Weisstein, Eric W., "Generalized Fermat Number", MathWorld.
- Yves Gallot, Generalized Fermat Prime Search
- Mark S. Manasse, Complete factorization of the ninth Fermat number (original announcement)
- Peyton Hayslette, Largest Known Generalized Fermat Prime Announcement


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DEFINITION of 'Usury Laws'
Regulations governing the amount of interest that can be charged on a loan. Usury laws specifically target the practice of charging excessively high rates on loans by setting caps on the maximum amount of interest that can be levied. These laws are designed to protect consumers.
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Optics InfoBase
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Projection operator method for biperiodic diffraction gratings with anisotropic/bianisotropic generalizations
Diffraction model of peristrophic multiplexing with spherical reference wave
Three-dimensional extended nonlocal photopolymerization driven diffusion model. Part II. Photopolymerization and model development
Recording and reconstruction of vector fields in a Fe-doped LiNbO3 crystal
Real-time 3D particle manipulation visualized using volume holographic gratings
Near-infrared sensitive photorefractive device using polymer dispersed liquid crystal and BSO:Ru hybrid structure
Nanoparticle polymer composite volume gratings incorporating chain transfer agents for holography and slow-neutron optics
Broadband angular filtering with a volume Bragg grating and a surface grating pair
Highly efficient volume hologram multiplexing in thick dye-doped jelly-like gelatin
Effects of chain-transferring thiol functionalities on the performance of nanoparticle-polymer composite volume gratings
Diffraction of volume Bragg gratings under high flux laser irradiation
Background suppression in long-distance imaging using volume hologram filters
Portable waveguide display system with a large field of view by integrating freeform elements and volume holograms
Wigner analysis of three dimensional pupil with finite lateral aperture
Effects of symbol modulation coding on readout fidelity of shift-multiplexed holographic digital data page storage in a photopolymerizable nanoparticle-(thiol-ene)polymer composite film
Design of high-resolution and multilevel reference pattern for improvement of both light utilization efficiency and signal-to-noise ratio in coaxial holographic data storage
Shift-multiplexed self-referential holographic data storage
Spectral interference in multiplexed volume Bragg gratings: theoretical calculations and experimental verification
Holographic characteristics of a modified photothermorefractive glass
Quantitative modeling of the reaction/diffusion kinetics of two-chemistry diffusive photopolymers
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