17.10.18

Self-sufficient exponents

Definition: A self sufficient exponent e is an exponent such that (me (mod n))e (mod n) = m for all m < n.

Conjecture: If n is a positive integer greater than 6, then there are at least 2 self-sufficient exponents smaller than lcm((p-1)(q-1)).

Example: Let n = 35, then 5, 7, 11 are self sufficient exponents because for all m < 35 (m^5)^5 = m, (m^7)^7 = m, (m^11)^11 = m. But also 17, 23, 29 are self-sufficient exponents.

Let n = 77, then 11, 19, 29 are self-sufficient exponents because for all m < 77 (m^11)^11 = m, (m^19)^19 = m, (m^29)^29 = m. But also 41, 49, 59 are also self-sufficient exponents.

Conjecture: If n is the product of two distinct odd primes p and q and k = ((lcm((p-1)(q-1))) - 1) then k is a self-sufficient exponent.

This has to do with the structure of the exponentiation table of Z sub n when n is the product of two distinct odd primes.

https://en.wikipedia.org/wiki/RSA_(cryptosystem), last retrieved 17/10/18