website shows around 100 zeroless pandigital Friedman numbers . Two of them are: 123456789 = ((86 + 2 × 7)5 − 91) / 34, and 987654321 = (8 × (97 + 6/2)5 + 1) / 34. Only one of them is nice: 268435179 = −268 + 4(3×5 − 17) − 9. Michael Brand proved that the density of Friedman numbers among the naturals is 1, which is to say that the probability of a number chosen randomly and uniformly between 1 and n to be a Friedman number tends to 1 as n tends to infinity. This result extends to Friedman numbers under any base of representation. He also proved that the same is true also for binary, ternary and quaternary nice Friedman numbers. The case of base-10 nice Friedman numbers is still open. Vampire numbers are a subset of Friedman numbers where the only operation is a multiplication of two numbers with the same number of digits, for example 1260 = 21 × 60. Finding 2-digit Friedman numbers There usually are fewer 2-digit Friedman numbers than 3-digit and more in any given base, but the 2-digit ones are easier to find. If we represent a 2-digit number as mb + n, where b is the base and m, n are integers from 0 to b−1, we need only check each possible combination of m and n against the equalities mb + n = mn, and mb + n = nm to see which ones are true. We need not concern ourselves with m + n or m × n, since these will always be smaller than mb + n when n < b. The same clearly holds for m − n and m / n. Other bases General results In base , is a Friedman number (written in base as 1mk = k × m1). In base , is a Friedman number (written in base as 100...00200...001 = 100..0012, with zeroes between each nonzero number). In base , is a Friedman number (written in base as 2k = k2). From the observation that all numbers of the form 2k × b2n can be written as k000...0002 with n 0's, we can find sequences of consecutive Friedman numbers which are arbitrarily long. For example, for , or in base 10, 250068 = 5002 + 68, from which we can easily deduce the range of consecutive Friedman numbers from 250000 to