Can anyone prove the following:
If $p = a\cdot b\cdot c$ is a Carmichael Number and $ m = \operatorname{lcm}((a-1), (b-1), (c-1))$ (least common multiple), prove $(p-1)/m$ is prime. (This applies to Carmichael Numbers with more factors.)
My research efforts show that the proof for this is the basis this primality test (a converse of Fermat's Little Theorem) but how the primality test and/or my claim are related I am not sure. Thanks for helping me on this.
The proposition is wrong, it can be disproved by counter examples: $$1729 = 7 \cdot 13 \cdot 19$$ $$m = \operatorname{lcm}(6, 12, 18) = 36$$ $$(p - 1) / m = 1728 / 36 = 48$$
$$41041 = 7 \cdot 11 \cdot 13 \cdot 41 $$ $$m = \operatorname{lcm}(6, 10, 12, 40) = 120$$ $$(p - 1) / m = 41040 / 120 = 242$$
Other counter examples are 2465, 8911, 15841, 62745 and (infinitely?) many more.