“Here is a purely computational problem (this problem cannot be attacked by any other means at present). Call a prime p good if every even number 2r £
p — 3 can be written in the form q1 - q2 where q1
£
p, q2 £
p are primes. Are there infinitely many good primes? The first bad prime is
97, I think. Selfridge and Blecksmith have tables of good primes up to 1037 at least, and they are surprisingly numerous.”
“I proved long ago that every m < n! is the distinct sum of n - 1 or fewer divisors of n!. Let h(m) be the smallest integer, if it exists, for which every integer less than m is the distinct sum of h(m) or fewer divisors of m. Srinivasan called the number for which h(m) exists practical. It is well known and easy to see that almost all numbers m are not practical. I conjured that there is a constant c £ 1 for which for infinitely many m we have h(m) < (loglogm)c. M Vose proved that h(n!) < cn1/2. Perhaps h(n!) < c (log n)c. I would be very glad to see a proof that h(n!) < ne.”
“A practical number m is called a champion if for every m > n, we have h(m) > h(n). For instance, 6 and 24 are champions, as h(6) = 2, the next practical number is 24, h(24) = 3, and for every m > 24, we have h(m) > 3. It would be of some interest to prove some results about champions. A table of the champions < 106 would be of some interest. I conjecture that n! is not a champion for n > n0.”