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,
2^{n} is never congruent to 1 (mod n)
2^{n} is congruent to 2 (mod n) whenever n is prime, and sometimes when it isn't,
is 2^{n} ever congruent to 3 (mod n)?
Here is the sequence of 2^n mod n.
0, 0, 2, 0, 2, 4, 2, 0, 8, 4, 2, 4, 2, 4, 8, 0, 2, 10, 2, 16, 8, 4, 2, 16, 7...
It might first seem like 2^n mod n is always either 0 or a power of 2. However, 2^18 mod 18=262144 mod 18=10, which is neither 0 nor a power of 2. Then, it might seem like 2^n mod n is always even. However, 2^25 mod 25=33554432 mod 25=7, which is odd. Then, it might seem like 2^n mod n is never 3. However, 2^4700063497 mod 4700063497=3.

Posted by Math Man
on 20171004 14:54:22 