New Mixed Proth Theorem

[sweety439]

See PrimeGrid New Sierpinski Problem and Mixed Sierpinski Theorem

For every odd k < 78557, there is an n such that 2^n > k and either k*2^n+1 or 2^n+k (or both) is prime

For the k*2^n+1 case, there are only 5 k's < 78557 remain with no known k with 2^n > k: {23971, 45323, 50777, 50873, 76877}, see http://boincvm.proxyma.ru:30080/test...ob_problem.php, together with the 5 remain k's in the original Sierpinski: {21181, 22699, 24737, 55459, 67607}, there are 10 remain k's

And there are known primes of the form 2^n+k with 2^n > k for these k:

2^28+21181
2^26+22699
2^11152+23971 (certificate)
2^17+24737
2^47+45323
2^61+50777
2^55+50873
2^746+55459
2^16389+67607 (certificate)
2^37+76877

And hence the new mixed Proth (Sierpinski) problem is now a theorem!!!

You can try the new mixed Riesel problem, i.e. for odd k < 509203, is there always an n such that 2^n > k and either k*2^n-1 or 2^n-k (or both) is prime?

[sweety439]

Originally Posted by sweety439

You can try the new mixed Riesel problem, i.e. for odd k < 509203, is there always an n such that 2^n > k and either k*2^n-1 or 2^n-k (or both) is prime?

For the Riesel side, see the attached file for odd k < 25000

For k = 2293, we already known that 2293*2^12918431-1 is prime, for k = 14347, we have a prime 14347*2^25997-1 and a PRP 2^130099-14347, thus we known that all odd k < 25000 have a prime, but can this "New Mixed Riesel Problem" become a theorem?