A115038 Smallest x such that x^2 + p is a perfect power over the set of primes p.

5, 1, 2, 1, 4, 6, 8, 9, 2, 14, 1, 18, 20, 21, 9, 26, 29, 8, 33, 21, 36, 7, 41, 6, 48, 50, 5, 53, 4, 56, 1, 65, 68, 69, 74, 19, 78, 81, 7, 86, 8, 90, 5, 96, 98, 12, 105, 17, 4, 114, 116, 2, 120, 44, 128, 131, 134, 27, 138, 140, 141, 146, 6, 155, 156, 158, 165, 168, 173, 174, 176, 37

Offset

1,1

Comments

Conjecture: There will always be an x,y,n such that x^2 + p = y^n for all primes p. In otherwords, there is a one-one mapping of the prime numbers to y^n - x^2 for some x,y,n. Therefore primes of the form y^n - x^2 are infinite in number.

Links

Table of n, a(n) for n=1..72.

Example

5 is the smallest number that when we add its square to prime 2, we get a
perfect power, 3^3. 1 is the smallest number that when we add its square to
prime 3, we get a perfect power, 2^2. So 5 and 1 are the first two entries in
the table.

Prog

(PARI) sqplusp(n) = { local(p, x, y, c=0); forprime(p=2, n, for(x=1, n, y=x^2+p; if(ispower(y), print1(x", "); c++; break ) ) ); print(); print(c) }

Crossrefs

Sequence in context: A265824 A097413 A202352 * A231990 A010131 A176323

Adjacent sequences: A115035 A115036 A115037 * A115039 A115040 A115041

Keyword

easy,nonn

Author

Cino Hilliard, Feb 26 2006

Status

approved