2¹ mod 1=2 mod 1=0
2² mod 2=4 mod 2=0
2³ mod 3=8 mod 3=2
2⁴ mod 4=16 mod 4=0
2⁵ mod 5=32 mod 5=2
2⁶ mod 6=64 mod 6=4
2⁷ mod 7=128 mod 7=2
2⁸ mod 8=256 mod 8=0
2⁹ mod 9=512 mod 9=8
2¹⁰ mod 10=1024 mod 10=4

Therefore, the sequence 2ⁿ mod n starts 0, 0, 2, 0, 2, 4, 2, 0, 8, 4, ... Notice that all of the terms so far are either 0 or a power of 2.

1. What is the smallest positive integer n such that 2ⁿ mod n is neither 0 nor a power of 2?
2. What is the smallest positive integer n such that 2ⁿ mod n is odd?
3. Is there a positive integer n such that 2ⁿ mod n=1?
4. Which positive integer from 2 to 10 is the last to appear as 2ⁿ mod n for the first time?

(In reply to Parts 1, 2, and 4 (and probably 3) by Larry)

OEIS A036236 smallest integer k > 0 such that 2^k mod k = n, or 0 if no such k exists1

has all such answers including for 1 and 3:

 1, 0, 3, 4700063497, 6, 19147, 10669, 25, 9, 2228071, 18, 262279, 3763, 95, 1010, 481, 20, 45, 35, 2873, 2951, 3175999, 42, 555, 50, 95921, 27, 174934013, 36, 777, 49, 140039, 56, 2463240427, 110, 477, 697, 91, 578, 623, 156, 2453, 540923, 55, 70, 345119, 287 

starts with entry for zero 2^1 mod 1 = 0. Then a zero for n=1, indicating there is no such k.

Edited on May 13, 2022, 11:47 am

Posted by Charlie on 2022-05-13 11:46:37