Before trying the problem "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."
For integers greater than 1,
2n is never congruent to 1 (mod n)
2n is congruent to 2 (mod n) whenever n is prime, and sometimes when it isn't,
is 2n ever congruent to 3 (mod n)?
My intuition was yes for two reasons
1) first, depending on your interpretation of 3 mod n when n<3, then n=1 could work since 3 mod 1 = 0 and 2^1 mod 1 = 0
2) I would expect 2^n mod n to be fairly chaotic so I would expect there would eventually be a value 3 mod n.
A quick search of OEIS and we are given http://oeis.org/A050259 which has the first value other than 1 (depending if you want to count it) is 4700063497
Posted by Daniel
on 2017-09-24 13:12:51