Cranky Numbers: From the 3rd grade to Fermat's Last Theorem

What's the connection between 3rd grade math and Fermat's Last Theorem? My 3rd grader comes home with problems that ask him how many ways he can write a number, such as 4, prompting him to list such expressions as: 1+3, 2 x 2, and 8÷2. If you restrict the list to sums of non-negative integers, the list is short and finite: 0+4, 1+3, 2+2, 1+1+2, 1+1+1+1. The number of sums in the list is called the partition number; the fourth partition number is thus 5. A patient 3rd grader (if such existed) could find the partition number of any integer. Partition numbers are handy if you are a 3rd grade teacher making up problems to occupy your students or a particle physicist.

If you looked at a list of partition numbers (the first 20 are: 1 2 3 5 7 11 15 22 30 42 56 77 101 135 176 231 297 385 490 627 more), you might notice that starting with the 4th partition number (5) every 5th number is divisible evenly by 5. Beginning with the 5th number (7), every 7th number divided evenly by 7. Perhaps not surprisingly, every 11th number after 11 is also divisible evenly by 11. The pattern ends there, but not the mystery. Indian mathematician Srinivasa Ramanujan recognized the patterns almost 100 years ago, but it took almost 40 years before Freeman Dyson explained them by inventing a function which he called the rank. Dyson's rank only explained the 5 and 7 patterns, roughly another 40 years would pass (is there a pattern here?) before the invention of the crank would account for the 11's.

As it turns out, there are more sequences buried in the list of partition numbers, you just have to know where to find them (a pattern based on those divisible by 13 begins with the 111,247th partition number). It also helps to have some of the techniques in number theory developed by Andrew Wiles to prove Fermat's Last Theorem. A proof that such patterns will exist for any prime number larger than 3 was published this year by Karl Mahlburg, a graduate student in math at the University of Wisconsin.