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Numbers k such that 2^k - 1 can be written in the form a^2 + 3*b^2.
+10
4
2, 3, 5, 6, 7, 9, 13, 14, 15, 17, 18, 19, 21, 25, 26, 27, 31, 37, 38, 39, 42, 45, 49, 51, 54, 57, 61, 62, 63, 65, 67, 74, 75, 78, 81, 85, 89, 93, 98, 101, 103, 107, 111, 114, 117, 122, 125, 126, 127, 133, 134, 135, 139, 147, 153, 162, 171, 183, 186, 189, 195, 201, 217, 221, 222, 225, 234, 243, 254, 255, 257, 259, 267, 269, 271, 278, 279, 281, 293, 294
COMMENTS
These 2^k - 1 numbers have no prime factors of the form 2 (mod 3) to an odd power.
EXAMPLE
2^67 - 1 = 10106743618^2 + 3*3891344499^2 = 9845359982^2 + 3*4108642899^2.
PROG
(PARI) for(i=2, 100, a=factorint(2^i-1)~; has=0; for(j=1, #a, if(a[1, j]%3==2&&a[2, j]%2==1, has=1; break)); if(has==0, print(i" -\t"a[1, ])))
EXTENSIONS
14 more terms from V. Raman, Aug 29 2012
Prime numbers p such that the Mersenne number 2^p - 1 can be written in the form a^2 + 3*b^2.
+10
3
2, 3, 5, 7, 13, 17, 19, 31, 37, 61, 67, 89, 101, 103, 107, 127, 139, 257, 269, 271, 281, 293, 347, 349, 353, 373, 379, 401, 457, 461, 499, 521, 523, 569, 577, 607, 631, 647, 727, 751, 863, 881, 907, 983, 1039, 1061, 1063, 1193
COMMENTS
These numbers have no prime factors congruent to 2 (mod 3) raised to an odd power. Prime factors which are == 2 (mod 3) come in pairs.
Mersenne exponents, A000043, are a proper subset.
EXAMPLE
2^67 - 1 = 10106743618^2 + 3*3891344499^2 = 9845359982^2 + 3*4108642899^2.
MATHEMATICA
fQ[n_] := Union[ Mod[ Transpose[ FactorInteger[2^n - 1]][[1]], 3]] == {1}; p = 2; lst = {}; While[p < 300, If[ fQ@ p, AppendTo[lst, p]; Print@ p]; p = NextPrime@ p] (* Or *)
p=2; (* open the first or second link and copy the listed factors for the prime exponent and paste into the parentheses that follow and change any periods to commas *) p = NextPrime@ p; pf = {}; Mod[Flatten[{pf, (2^p - 1)/Times @@ pf}], 3] (* Robert G. Wilson v, Aug 26 2012 *)
PROG
(PARI) forprime(i=2, 100, a=factorint(2^i-1)~; has=0; for(j=1, #a, if(a[1, j]%3==2&&a[2, j]%2==1, has=1; break)); if(has==0, print(i" -\t"a[1, ])))
Even numbers n such that 2^n - 1 can be written in the form a^2 + 3*b^2.
+10
3
2, 6, 14, 18, 26, 38, 42, 54, 62, 74, 78, 98, 114, 122, 126, 134, 162, 186, 222, 234, 254, 278, 294, 342, 366, 378, 402, 434, 486, 518, 558, 666, 702, 762, 834, 882, 914, 1026, 1098, 1134, 1206, 1302, 1458, 1554, 1674, 1998, 2106
COMMENTS
These 2^n-1 numbers have no prime factors of the form 2 (mod 3) to an odd power.
EXAMPLE
2^67-1 = 10106743618^2+3*3891344499^2 = 9845359982^2+3*4108642899^2
MATHEMATICA
Select[Range[2, 200, 2], Length[FindInstance[x^2 + 3*y^2 == 2^# - 1, {x, y}, Integers]] > 0 &] (* G. C. Greubel, Apr 14 2017 *)
PROG
(PARI) for(i=2, 100, a=factorint(2^i-1)~; has=0; for(j=1, #a, if(a[1, j]%3==2&&a[2, j]%2==1, has=1; break)); if(has==0&&i%2==0, print(i" -\t"a[1, ])))
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