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Old 2016-12-07, 15:01   #1
sweety439
 
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Nov 2016

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Default Dual Sierpinski/Riesel prime

There is a "Five or bust!" project, to search the primes of the form 2^n+k for all odd numbers k<78557, this is because 2^n+k is the dual of k*2^n+1. For k*2^n-1, the dual of it is |2^n-k|.

The project found the probable primes 2^1518191+75353, 2^2249255+28433, 2^4583176+2131, 2^5146295+41693, and 2^9092392+40291.

For generalized Sierpinski/Riesel conjecture to base b, the dual of k*b^n+1 is (b^n+k)/gcd(b^n, k), and the dual of k*b^n-1 is (|b^n-k|)/gcd(b^n, k).

These are the dual (probable) primes that I found of the Sierpinski/Riesel conjectures that has only one k remaining for bases b<=144: (See http://mersenneforum.org/showpost.ph...42&postcount=1)

form dual form least prime for the "dual form" dual n
2036*9^n+1 9^n+2036 9^4+2036 4
7666*10^n+1 (10^n+7666)/2 (10^67+7666)/2 67
244*17^n+1 17^n+244 17^838+244 838
5128*22^n+1 (22^n+5128)/8 (22^11+5128)/8 11
398*27^n+1 27^n+398 27^7+398 7
166*43^n+1 43^n+166 ? (>1000)
17*68^n+1 (68^n+17)/17 (68^1+17)/17 1
1312*75^n+1 75^n+1312 ? (>1000)
8*86^n+1 (86^n+8)/8 (86^205+8)/8 205
32*87^n+1 87^n+32 ? (>1000)
1696*112^n+1 (112^n+1696)/32 (112^44+1696)/32 44
48*118^n+1 (118^n+48)/16 (118^57+48)/16 57
34*122^n+1 (122^n+34)/2 (122^2+34)/2 2
40*128^n+1 (128^n+40)/8 (128^2+40)/8 2
form dual form least prime for the "dual form" dual n
1597*6^n-1 |6^n-1597| |6^3-1597| 3
4421*10^n-1 |10^n-4421| |10^212-4421| 212
3656*22^n-1 (|22^n-3656|)/8 ? (>1000)
404*23^n-1 |23^n-404| |23^568-404| 568
706*27^n-1 |27^n-706| |27^2-706| 2
424*93^n-1 |93^n-424| |93^1-424| 1
29*94^n-1 |94^n-29| |94^2-29| 2
924*103^n-1 |103^n-924| |103^1-924| 1
84*109^n-1 |109^n-84| |109^6-84| 6
24*123^n-1 (|123^n-24|)/3 (|123^5-24|)/3 5
926*133^n-1 |133^n-926| |133^2-926| 2

Last fiddled with by sweety439 on 2016-12-07 at 15:03
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