A239711 Twin primes of the form m = b^i + b^j +- 1, where i > j > 0, b > 1.

5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 71, 73, 107, 109, 149, 151, 191, 193, 239, 241, 269, 271, 419, 421, 461, 463, 599, 601, 809, 811, 1031, 1033, 1151, 1153, 1301, 1303, 1451, 1453, 1481, 1483, 1721, 1723, 1871, 1873, 2111, 2113, 2267, 2269, 2549, 2551, 2969, 2971, 3389, 3391, 3539, 3541

Offset: 1

Author

Hieronymus Fischer, Mar 27 2014 and May 04 2014

Keywords

• nonn

Comments

(a(2k-1), a(2k)), k > 0, form pairs of twin primes.

Numbers m that satisfy m = b^i + b^j + 1 and b == 1 (mod 3) and those that satisfy m = b^i + b^j - 1 with odd i and j and b == 2 (mod 3) are never terms, since they are divisible by 3. It follows that no numbers 4^i + 4^j +- 1, or 7^i + 7^j +- 1, or 10^i + 10^j +- 1, ... can be terms. Also, no numbers 5^(2m-1) + 5^(2k-1) +- 1, or 8^(2m-1) + 8^(2k-1) +- 1, or 11^(2m-1) + 11^(2k-1) +- 1, ... with m > k > 0, can be terms.

Example 1: 10^6 + 10^4 + 1 = 1010001 is not a term, since 10 == 1 (mod 3); certainly, 1010001 = 3*336667.

Example 2: 8^9 + 8^7 - 1 = 136314879 is not a term, since 8 == 2 (mod 3) and i, j odd; certainly 136314879 = 3*45438293.

Examples

			a(1) = 5, since 5 = 2^2 + 2^1 - 1 is prime.
a(2) = 7, since 7 = 2^3 + 2^1 + 1 is prime.
a(7) = 29, since 29 = 3^3 + 3^1 - 1 is prime.
a(8) = 31, since 31 = 3^3 + 3^1 + 1 is prime.
a(9) = 41.
a(10) = 43.
a(99) = 43889.
a(100) = 43891.
a(999) = 233524241.
a(1000) = 233524243.
a(9999) = 110211052379.
a(10000) = 110211052381.
a(99999) = 27208914574871.
a(100000) = 27208914574873.
a(199999) = 136140088764371.
a(200000) = 136140088764373.
[the last two terms form the 100000th twin prime pair of the form b^i + b^j +-1]
		

Links

• Hieronymus Fischer, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A239709, A239710, A239712 - A239720.