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A090738 Integers n such that the concatenation of n^2 and (n+1)^2 is prime.

%I A090738 #15 Dec 23 2018 12:48:46
%S A090738 8,12,18,20,28,40,48,82,88,92,96,98,112,128,132,140,142,218,232,238,
%T A090738 240,246,252,272,286,288,330,332,346,356,360,376,380,450,458,460,462,
%U A090738 466,488,500,518,532,538,550,588,590,596,602,610,612,616,630,640,646,648
%N A090738 Integers n such that the concatenation of n^2 and (n+1)^2 is prime.
%C A090738 I conjecture this sequence to be infinite. Searching through the first 200000 values, I found 7000 primes, of which over 400 were "twins", i.e. both n^2*(n+1)^2 and (n+2)^2*(n+3)^2 were prime, where "*" denotes concatenation. I conjecture there to be an infinitude of such twins and the obvious generalizations.
%C A090738 The symmetric problem, i.e., finding two consecutive primes whose concatenation is a square, is somehow harder. Probably the smallest such primes are p = 411828016678198512725064549221 and its successor p+20, whose concatenation is equal to 641738277398347583345401533579^2. - _Giovanni Resta_, Jul 23 2015
%H A090738 Zak Seidov, <a href="/A090738/b090738.txt">Table of n, a(n) for n = 1..10000</a>
%e A090738 The first term, n=8 corresponds to the prime 6481, which is the concatenation of 8^2=64 and 9^2=81. The second term, n=12 corresponds to the prime 144169.
%t A090738 For[i=1, i<200000, i=i+1, n=2i;e=IntegerPart[2 Log[10, n+1]]+1;x=10^e n^2 + (n+1)^2;y={n, x}; If[ PrimeQ[x], Save["primes.txt", y]]]
%t A090738 Select[Range@ 648, PrimeQ@ FromDigits[IntegerDigits[#^2] ~Join~ IntegerDigits[(# + 1)^2]] &] (* _Michael De Vlieger_, Jul 23 2015 *)
%t A090738 Position[FromDigits[Flatten[IntegerDigits/@#]]&/@Partition[ Range[ 700]^2, 2,1],_?PrimeQ]//Flatten (* _Harvey P. Dale_, Dec 23 2018 *)
%Y A090738 See A104242 for the corresponding primes.
%K A090738 nonn,base
%O A090738 1,1
%A A090738 Alex Kontorovich (alexk(AT)math.columbia.edu), Jan 19 2004