Sophie Germain's theorem for prime pairs p, 6p + 1

AbstractIn 1823, Sophie Germain (Mem. Acad. Sci. Inst. France 6 (1823), 1–60) showed that if p, 2p + 1 are both odd primes then the so-called “First Case” of Fermat's Last Theorem holds for p. This was extended by Legendre, Wendt, Vandiver, Denes, and others to prime pairs p, mp + 1, where 6 ∤ m and p is sufficiently large (depending on m). The cases where 6 divides m are fraught with an inescapable technical difficulty, and, as we shall see in this paper, it requires quite sophisticated techniques to even find a partial resolution for prime pairs p, 6p + 1

Flat primes and thin primes

A number is called upper (lower) flat if its shift by +1 ( −1) is a power of 2 times a squarefree number. If the squarefree number is 1 or a single odd prime then the original number is called upper (lower) thin. Upper flat numbers which are primes arise in the study of multi-perfect numbers. Here we show that the lower or upper flat primes have asymptotic density relative to that of the full set of primes given by twice Artin’s constant, that more than 53% of the primes are both lower and upper flat, and that the series of reciprocals of the lower or the upper thin primes converges