On the Exclusion of Carmichael Numbers Among Certain Mersenne Numbers M_p with p = 4r + 1

Abstract. Let p = 4r + 1 with p and r prime, and let Mₚ = 2ᵖ − 1. For every odd m ≥ 3 we prove an explicit upper bound: if p > 2 · 31ᵐ − 61, then Mₚ is not a Carmichael number with m prime factors. More precisely, a Carmichael Mₚ with m prime factors must have p in an explicit finite set Aₘ obtained from simple congruences. Excluding each exponent in Aₘ by an exact certificate, we prove that for m = 3, 5, 7 and 9 no such Mₚ is a Carmichael number with m prime factors. For every larger odd m, the question reduces to the finitely many exponents in Aₘ.