posts by S.M.R. on groups.io/PrimeNumbersTheory
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2026-05-17: “This is prime for all n ≥ 4: Mod[(2 n - 2)!!^4, 2 n] + 3 Mod[(n - 1)! + 1, n] - 1”
Yes, as the list clearly shows, it’s equal to n if n is prime (due to Wilson’s and Fermat’s Little Thm,
$(p-1)! = -1 \pmod p$ and $2^p = 2 \pmod p$), and equal to 0 + 3(0+1) -1 = 2 else. -
2026-05-14: “If q[n] = 2 n Floor[n^(1/GoldenRatio)] + 1 divides 2^n+1, then q[n] is prime”
(PARI) phi=(sqrt(5)-1)/2; for(n=1,oo, Mod(2, q=2*n*floor(n^phi)+1)^n+1 || isprime(q)|| print1(n", "))
Note: floor(n^phi) = floor(exp phi ln n) -
2026-05-10: “If 2 n Floor[Sqrt[n]] + 1 divides 2^n+1, then 2 n Floor[Sqrt[n]] + 1 is prime”
counter-examples: 167990, 3907750, 50806483, 70709130, …
(PARI) for(n=1,oo, Mod(2, q=2*n*sqrtint(n)+1)^n+1 || isprime(q)|| print1(n", ")) /* 0.02 sec for 167990 */ -
2026-04-25: “p is prime if and only if (p-2)!!^4=1 (mod p)”
counter-examples: ?
(PARI) DoubleFactMod(n, m)=prod(k=1,n\2-1, n-2*k, Mod(n,m))\ for(n=1,oo, lift(DoubleFactMod(p-2, p)^4)==1==isprime(n)|| print1(n", "))