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

%I A115431 #13 Feb 20 2019 15:01:34
%S A115431 6,5346,8083,10578,45531,58626,2392902,2609443,7272838,51248898,
%T A115431 98009803,159728062051,360408196038,523637103531,770378933826,
%U A115431 998000998003,1214959556998,1434212848998,3860012299771,4243705560771
%N A115431 Numbers k such that the concatenation of k with k-2 gives a square.
%C A115431 From _Robert Israel_, Feb 20 2019: (Start) The same as A116117 and A116135 (see link).
%C A115431 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.
%C A115431 For each k >= 1, 10^(4*k)-2*10^(3*k)+10^(2*k)-2*10^k+3 is a term.
%C A115431 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)
%H A115431 Robert Israel, <a href="/A115431/b115431.txt">Table of n, a(n) for n = 1..1312</a>
%H A115431 Robert Israel, <a href="/A115431/a115431.pdf">Proof that A115431, A116117 and A116135 are the same</a>
%e A115431 8083_8081 = 8991^2.
%e A115431 98009803_98009800 = 98999900 * 98999902, where _ denotes
%e A115431 concatenation
%p A115431 f:= proc(n) local S;
%p A115431   S:= map(t -> rhs(op(t))^2 mod 10^n+2, [msolve(x^2+2,10^n+1)]);
%p A115431   op(sort(select(t -> t-2 >= 10^(n-1) and t-2 < 10^n and issqr(t-2 + t*10^n), S)))
%p A115431 end proc:
%p A115431 seq(f(n),n=1..20); # _Robert Israel_, Feb 20 2019
%Y A115431 Cf. A030465, A102567, A115426, A115437, A115428, A115429, A115430, A115432, A115433, A115434, A115435, A115436, A115442.
%K A115431 base,nonn
%O A115431 1,1
%A A115431 _Giovanni Resta_, Jan 24 2006