Concatenation of two Mersenne numbers
 

Concatenate 2^k-1 and 2^(k-2)-1. Example 71, 153,...can you find a prime congruent to 4 or 3 mod 7?

:sleep:

Here's the remainders of the first few numbers of that form:

[code]71 mod 7 = 1
153 mod 7 = 6
317 mod 7 = 2
6315 mod 7 = 1
12731 mod 7 = 5
25563 mod 7 = 6
511127 mod 7 = 1
1023255 mod 7 = 2
2047511 mod 7 = 4
40951023 mod 7 = 1
81912047 mod 7 = 0
163834095 mod 7 = 5
327678191 mod 7 = 1
6553516383 mod 7 = 1
13107132767 mod 7 = 1
26214365535 mod 7 = 1
524287131071 mod 7 = 4
[/code]Out of those 17 numbers, only 71 and 317 are prime. It seems that the only possible remainders are 0, 1, 2, 4, 5, and 6.

the possible remainders are 0,1,2,3,4,5,6 (3 with frequency 1/18)
Can you find a prime with residue 3 or 4?

PRPs:
[CODE]M(3)M(1) 1
M(5)M(3) 2
M(25)M(23) 6
M(33)M(31) 1
M(143)M(141) 2
M(165)M(163) 1
M(285)M(283) 1
M(321)M(319) 1
M(323)M(321) 2
M(1003)M(1001) 5
M(1683)M(1681) 1
M(1921)M(1919) 5
M(4223)M(4221) 1
M(5451)M(5449) 1
M(6783)M(6781) 1
[/CODE]
The number following M(x)M(x-2) is the number mod 7