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A115431 Numbers k such that the concatenation of k with k-2 gives a square. 22
6, 5346, 8083, 10578, 45531, 58626, 2392902, 2609443, 7272838, 51248898, 98009803, 159728062051, 360408196038, 523637103531, 770378933826, 998000998003, 1214959556998, 1434212848998, 3860012299771, 4243705560771 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
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)
LINKS
EXAMPLE
8083_8081 = 8991^2.
98009803_98009800 = 98999900 * 98999902, where _ denotes
concatenation
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
CROSSREFS
Sequence in context: A066061 A028366 A305886 * A116117 A116135 A367958
KEYWORD
base,nonn
AUTHOR
Giovanni Resta, Jan 24 2006
STATUS
approved

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Last modified April 20 15:37 EDT 2024. Contains 371844 sequences. (Running on oeis4.)