Problems & Puzzles: Puzzles

Puzzle 1218 3, 41, 463, ... Giorgos Kalogeropoulos sent the following nice puzzle: Primes p = 3,41 and 463 have the following cyclic property: q=prime(p) and the Largest Prime Factor (LPF) of prime(q)+prime(q+1) is p For p=3, we have q=prime(p)=prime(3)=5 LPF(prime(q)+prime(q+1)) = LPF(prime(5)+prime(6)) = LPF(11+13) = LPF(24) = LPF(2*2*2*3) = 3 = p For p=41 we have q=prime(41)=179 LPF(prime(179)+Prime(180)) = LPF(1063+1069) = LPF(2132) = LPF(2*2*13*41) = 41 = p The same goes for p=463 -> q=3299 -> LPF(30557+30559) = LPF(2*2*3*11*463) = 463 = p Q. Can you find other primes with the same property or prove they don't exist ?

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On May 3, 2025, Oscar Volpatti wrote:

I've checked all primes up to p = prime(10^9) = 22801763489, finding one more solution:

  

n = 222617381,

p = prime(n) = 4724804933,

q = prime(p) = 115412584783,

x = prime(q) = 3203417744519,

y = prime(q+1) = 3203417744629,

z = x+y = 6406835489148,

z = 2^2*3*113*4724804933,

LPF(z) = p.

I've used programs primesieve (for fast sieving) and primecount (for double-checking) by Kim Walisch.

 

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