hence , this yields the well-known cannonball-stacking problem of Lucas, a fact which is connected to the Leech lattice. In addition, if in a Pythagorean -tuple () all addends are consecutive except one, one can use the equation, Since the second power of cancels out, this is only linear and easily solved for as though , should be chosen so that is an integer, with a small example being , yielding, Thus, one way of generating Pythagorean -tuples is by using, for various , where q = n–2 and where Fermat's Last Theorem A generalization of the concept of Pythagorean triples is the search for triples of positive integers , , and , such that , for some strictly greater than 2. Pierre de Fermat in 1637 claimed that no such triple exists, a claim that came to be known as Fermat's Last Theorem because it took longer than any other conjecture by Fermat to be proved or disproved. The first proof was given by Andrew Wiles in 1994. or th powers summing to an th power Another generalization is searching for sequences of positive integers for which the th power of the last is the sum of the th powers of the previous terms. The smallest sequences for known values of are: = 3: {3, 4, 5; 6}. = 4: {30, 120, 272, 315; 353} = 5: {19, 43, 46, 47, 67; 72} = 7: {127, 258, 266, 413, 430, 439, 525; 568} = 8: {90, 223, 478, 524, 748, 1088, 1190, 1324; 1409} For the case, in which called the Fermat cubic, a general formula exists giving all solutions. A slightly different generalization allows the sum of th powers to equal the sum of th powers. For example: (): 1 + 12 = 9 + 10, made famous by Hardy's recollection of a conversation with Ramanujan about the number 1729 being the smallest number that can be expressed as a sum of two cubes in two distinct ways. There can also exist positive integers whose th powers sum to an th power (though, by Fermat's Last Theorem, not for ; these are counterexamples to Euler's sum of powers conjecture. The smallest known counterexamples are : (95800, 217519, 414560; 422481) : (27, 84, 110, 133; 144) Heronian triangle triples A Heronian triangle is commonly defined as one with integer sides whose area is also an integer. The lengths of