A115431 Numbers k such that the concatenation of k with k-2 gives a square.

6, 5346, 8083, 10578, 45531, 58626, 2392902, 2609443, 7272838, 51248898, 98009803, 159728062051, 360408196038, 523637103531, 770378933826, 998000998003, 1214959556998, 1434212848998, 3860012299771, 4243705560771

Offset: 1

Author

Giovanni Resta, Jan 24 2006

Keywords

• base
• nonn

Comments

From Robert Israel, Feb 20 2019: (Start) The same as A116117 and A116135 (see link).

So there are two equivalent definitions: numbers k such that k concatenated with k-6 gives the product of two numbers which differ by 4; and numbers k such that k concatenated with k-3 gives the product of two numbers which differ by 2.

For each k >= 1, 10^(4*k)-2*10^(3*k)+10^(2*k)-2*10^k+3 is a term.

If k is a term and k-2 has length m, then all prime factors of 10^m+1 must be congruent to 1 or 3 (mod 8). In particular, we can't have m == 2 (mod 4) or m == 3 (mod 6), as in those cases 10^m+1 would be divisible by 101 or 7 respectively. (End)

Examples

			8083_8081 = 8991^2.
98009803_98009800 = 98999900 * 98999902, where _ denotes
concatenation
		

Links

• Robert Israel, Table of n, a(n) for n = 1..1312
• Robert Israel, Proof that A115431, A116117 and A116135 are the same

Crossrefs

Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115432, A115433, A115434, A115435, A115436, A115442.

Programs

Maple

f:= proc(n) local S;
  S:= map(t -> rhs(op(t))^2 mod 10^n+2, [msolve(x^2+2,10^n+1)]);
  op(sort(select(t -> t-2 >= 10^(n-1) and t-2 < 10^n and issqr(t-2 + t*10^n), S)))
end proc:
seq(f(n),n=1..20); # Robert Israel, Feb 20 2019