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A routine for testing a conjecture

The ec numbers are so defined: ec(k)=(2^k-1)10^d+2^(k-1)-1, where d is the number of decimal digits of 2^(k-1)-1. In other words these numbers are formed by the base 10 concatenation of two consecutive Mersenne numbers, for example: 157, 12763, 40952047... For some values of k, ec(k) is prime. I found that up to k=565.000 there is no prime of the form (2^k-1)10^d+2^(k-1)-1 which is congruent to 6 mod 7. So I conjectured that there is no prime of this form congruent to 6 mod 7. Has somebody an efficient program for Sage to test this conjecture further?

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A routine for testing a conjecture

The ec numbers are so defined: ec(k)=(2^k-1)10^d+2^(k-1)-1, defined:

ec(k) = (2^k-1)*10^d + 2^(k-1) - 1


where d d is the number of decimal digits of 2^(k-1)-1. 2^(k-1) - 1 . In other words these numbers are formed by the base 10 concatenation of two consecutive Mersenne numbers, for example: 157, 12763, 40952047... 40952047...

For some values of k, ec(k) k, ec(k) is prime. I found that up to k=565.000 there is no prime of the form (2^k-1)10^d+2^(k-1)-1 (2^k-1)*10^d + 2^(k-1) - 1 which is congruent to 6 mod 7. So I conjectured that there is no prime of this form congruent to 6 mod 7. Has somebody an efficient program for Sage to test this conjecture further?

A routine for testing a conjecture

The ec numbers are so defined:

ec(k) = (2^k-1)*10^d + 2^(k-1) - 1


where d is the number of decimal digits of 2^(k-1) - 1 . In other words these numbers are formed by the base 10 concatenation of two consecutive Mersenne numbers, for example: 157, 12763, 40952047...

For some values of k, ec(k) is probable prime. I found that up to k=565.000 there is no probable prime of the form (2^k-1)*10^d + 2^(k-1) - 1 which is congruent to 6 mod 7. So I conjectured that there is no probable prime of this form congruent to 6 mod 7. Has somebody an efficient program for Sage to test this conjecture further?