Problems & Puzzles:
Pseudo twin primes
Martín Ruiz sent the following puzzle:
positive integres greater than one.
n and m are pseudotwinprimes iff TW(n,m) is integer
that: If n,m are twin primes then n,m are pseudotwin primes.
examples of pseudotwinprimes (n, m) are:
(7,191) and (41, 1993). Find
more pairs of pseudotwinprimes that don’t are twin primes
Contributions cam from Farideh Firoozbakht, David Broadhurst & Alexey
1. TW(n,n+2) = ((n-1)!+1)((n+1)!+1)(n^2+(n+2)^2)/(n^2(n+2)^2+2n(n+2))
Now if both numbers n & n+2 are primes then by Wilson's Theorem
both numbers ((n-1)!+1)/n & ((n+1)!+1)/(n+2) are integers so if n &
n+2 are twin primes then n & n+2 are pseudotwin primes.
2. (59,13537) is a pair of pseudotwin primes that they aren't twin
Answer. Let n, m be primes, then we can rewrite TW(n,m) as follows
now, the two leader factors are integers by the Wilson's theorem.
Moreover, we can use m=n+2 (they are twin prime!), and
the last factor
TW(n,m) is an integer. QED.