A187868 Primes of the form 100^k - 10^k + 1.

9901, 99990001, 999999000001, 9999999900000001

Offset

1,1

Comments

These primes correspond to k = 2, 4, 6, and 8. There are no other primes of this form for k up to 10000. These primes appear in A187614 because the decimal representation of their reciprocal contains only the digits 0, 1, 8, and 9.

Because these numbers are values of the sixth cyclotomic polynomial, by Theorem 1 of Golomb, the only k that can produce primes are the 3-smooth numbers, 2^i*3^j (A003586).

Next term has k > 1000000. - Robert Gelhar, Aug 20 2020

References

A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, p. 85.

Links

Table of n, a(n) for n=1..4.

Solomon W. Golomb, Cyclotomic polynomials and factorization theorems, Amer. Math. Monthly 85 (1978), 734-737.

Mathematica

Select[Table[100^n - 10^n + 1, {n, 1000}], PrimeQ]

Crossrefs

Cf. A003586, A168624, A187614.

Sequence in context: A237917 A205612 A205350 * A072848 A145381 A212401

Adjacent sequences: A187865 A187866 A187867 * A187869 A187870 A187871

Keyword

nonn,more

Author

T. D. Noe, Mar 14 2011

Status

approved