D. If the prime can evenly divide one of these numerators evenly, it is an "irregular prime"
E. 37 divides the numerator of B(32) = - 7709321041217 evenly, so it is imperfect. 691 is also imperfect.
There is some special rule that if the prime has to be bigger than the Bernoulli number or it doesn't count. Which is why 5 & 7 don't count. I.e., 5 is not bigger than its B-number of 10; 7 isn't bigger than its B-number of 14 (Actually the rule is: bigger than 2 time the B value plus one, but that is getting a bit detailed)
The above steps will find some but not all imperfect primes - since Bernoulli numbers aren't the only "p-th cyclotomic field" type numbers.
I'm about 89% sure the above is correct. It's been a long long time.
bear with me, it's been a while
A. these things called "Bernoulli numbers" exist and their values can be calculated.
B. Bernoulli numbers have a numerator (and a denominator, but the denominator isn't important in this discussion).
C. Someone's done the math: on this page the second column labeled a(n) is a list of the first 40 numerators . http://www.research.att.com/~njas/sequences/table?a=27641&fmt=4
D. If the prime can evenly divide one of these numerators evenly, it is an "irregular prime"
E. 37 divides the numerator of B(32) = - 7709321041217 evenly, so it is imperfect. 691 is also imperfect.
There is some special rule that if the prime has to be bigger than the Bernoulli number or it doesn't count. Which is why 5 & 7 don't count. I.e., 5 is not bigger than its B-number of 10; 7 isn't bigger than its B-number of 14 (Actually the rule is: bigger than 2 time the B value plus one, but that is getting a bit detailed)
The above steps will find some but not all imperfect primes - since Bernoulli numbers aren't the only "p-th cyclotomic field" type numbers.
I'm about 89% sure the above is correct. It's been a long long time.