A168147 Primes of the form 10*n^3 + 1.

11, 271, 641, 2161, 33751, 40961, 58321, 138241, 196831, 270001, 297911, 466561, 506531, 795071, 1326511, 1406081, 1851931, 2160001, 3890171, 4218751, 5314411, 5513681, 6585031, 7290001, 8043571, 11910161, 12597121, 12950291, 14815441

Offset: 1

Author

Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 19 2009

Keywords

• nonn
• base
• easy

Comments

(1) These primes all with end digit 1=1^3 are concatenations of two CUBIC numbers: "n^3 1".

(2) It is conjectured that the sequence is infinite.

(3) It is an open problem if 3 consecutive naturals n exist which give such a prime.

No three such integers exist, as every n = 2 (mod 3) yields 10n^3 + 1 = 0 (mod 3). - Charles R Greathouse IV, Apr 24 2010

References

• Harold Davenport, Multiplicative Number Theory, Springer-Verlag New-York 1980
• Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications 2005

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A030430 (primes of the form 10*n+1).

Cf. A167535 (concatenation of two square numbers which give a prime).

See A168219 for the numbers n.

Programs

Magma

[ a: n in [1..150] | IsPrime(a) where a is 10*n^3+1 ]; // Vincenzo Librandi, Jul 25 2011

Mathematica

Select[Table[10*n^3+1,{n,1000}],PrimeQ] (* Vincenzo Librandi, Aug 01 2012 *)

PARI

for(n=1,2e2, isprime(n^3*10+1) && print1(n^3*10+1", "))  \\ M. F. Hasler, Jul 24 2011

Formula

a(n) = 10*A168219(n)^3 + 1. \\ M. F. Hasler, Jul 24 2011