The third and fourth Lev–Sonn primes

Abstract

Kalmynin calls a prime p a Lev–Sonn prime if p=2α(α-1)+1 and C(α²-1, n-1+α) ≡ (-1)ⁿ⁻¹C(α²-1, α) (mod p) for some n with 1<n ≤ α, and asks whether there are infinitely many. His search over α ≤ 3000 found two, p=13 and p=41. We exhibit the next two, p=25 293 378 613quad(α=112 458),qquad p=74 857 799 381quad(α=193 466), and prove that they are the third and the fourth: for every α with 2 ≤ α ≤ 300 000 the number 2α(α-1)+1 is a Lev–Sonn prime if and only if α ∈ {3,5,112 458,193 466}. The congruence witnessing the first of the two new primes relates binomial coefficients with 616 862 and 991 481 decimal digits, and the one witnessing the second relates larger ones; the tool that makes them decidable is an exact telescoping identity whose correction factors are units modulo p, and which replaces the pair of binomial coefficients by two products of n-1 linear factors. A substitution in it removes α from one side entirely and turns the condition into (2j+1)!! ≡ 2ʲ(α+j)!/α! (mod p) with j=n-1. We also correct the witness printed for p=41: the pair (α,n)=(5,5) recorded in the source is not a witness, (5,4) is, and it is the only one. Every theorem, proposition and lemma below is machine-checked in Lean 4.

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