[qtkyfhwo] Bracket primes

The following numbers trivially (because of their algebraic form x²-y²=(x+y)(x-y)) factor into two primes of nearly equal size surrounding a power of two. They are RSA-like, but would be terrible for RSA because they are easily factored by Fermat's factorization method.

2² - 0²
2⁴ - 1²
2⁸ - 3²
2¹⁶ - 15²
2³² - 15²
2⁶⁴ - 267²
2¹²⁸ - 2253²
2²⁵⁶ - 5823²
2⁵¹² - 357²
2¹⁰²⁴ - 169575²
2²⁰⁴⁸ - 151137²
2⁴⁰⁹⁶ - 2061267²
2⁸¹⁹² - 7542063²

f(mid)=for(i=0, 10^20, if(ispseudoprime(mid-i) && ispseudoprime(mid+i), answer=i; break)); answer

Sieve methods would tremendously improve performance in finding more bracket primes.

The 256-bit entry takes 8 minutes 32 seconds to factor with Pari/GP, including the futile ECM.

10⁴⁵±9 takes 5 to 8 hours to factor with the same default settings. What a difference 43 bits makes!

A047160 . By Goldbach's Conjecture, there exist bracket primes around any power of two.