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

8, 12, 18, 20, 28, 40, 48, 82, 88, 92, 96, 98, 112, 128, 132, 140, 142, 218, 232, 238, 240, 246, 252, 272, 286, 288, 330, 332, 346, 356, 360, 376, 380, 450, 458, 460, 462, 466, 488, 500, 518, 532, 538, 550, 588, 590, 596, 602, 610, 612, 616, 630, 640, 646, 648

Offset: 1

Author

Alex Kontorovich (alexk(AT)math.columbia.edu), Jan 19 2004

Keywords

• nonn
• base

Comments

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.

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

Examples

			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.
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..10000

Crossrefs

See A104242 for the corresponding primes.

Programs

Mathematica

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]]]
Select[Range@ 648, PrimeQ@ FromDigits[IntegerDigits[#^2] ~Join~ IntegerDigits[(# + 1)^2]] &] (* Michael De Vlieger, Jul 23 2015 *)
Position[FromDigits[Flatten[IntegerDigits/@#]]&/@Partition[ Range[ 700]^2, 2,1],?PrimeQ]//Flatten (* _Harvey P. Dale, Dec 23 2018 *)