Problems & Puzzles: Puzzles

Puzzle 648. P^2 = A^2&B^2 This week I started seeking for are squared primes that could be expressed as a concatenation of two squares. I found 18 of these primes less than 2^32: 7, 13, 19, 41, 223, 487, 1201, 1301, 12491, 18493, 32009, 110237, 282001, 1039681, 2380531, 3905413, 96799849, 1012639687, ... Examples (the smallest and largest): 7^2=4&9 = 2^2&3^2 1012639687^2 = 102543913568745796&9 = 320224786^2&3^2 But I was unable to find a squared prime expressible as a concatenation of two squares in two ways. Q1. For sure there are more solutions like these above. Send only your largest one solution. Q2. Find the smallest squared prime that is expressible as a concatenation of two squares in two or more ways.

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Contributions came from Hakan Summakoğlu, Emmanuel Vantieghem, Giovanni Resta,

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Hakan wrote:

Q1: I found a family with infinite terms that satisfy, for m=1,2,3,...

4(9)_m(0)_m1 ^2 = 24(9)_m-1 (0)_m 1 (9)_m 8(0)_m 1 = 4(9)_m ^2 & (9)_m+1 ^2

I tested for prime solutions and I found 4 primes. Prime solutions (for m<100) : m=9, 11, 23 and 47.

(Note1: Hakan missed one smaller prime solution -for m=4- the one gotten by Emanuel below)
(Note2: what is the next prime solution for the Hakan's model?)

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Emmanuel wrote:

I could add one more solution: 4999900001^2 = 24999000019999800001 = (49999^2)&(99999^2).

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Giovanni wrote:

Q2. I searched all the primes up to 6.25*10^12 but I found no prime whose square has two expressions, apart from the degenerate case
19976430247^2 = 39905776541325648100 | 9
19976430247^2 = 399057765413256481 | 009.

Q1. My largest (certified) prime has 26280 digits.

I noticed that
4999900001^2 = 2499900001&9999800001
with 2499900001 = 49999^2
and  9999800001 = 99999^2.

Generalizing, I found a prime P =  499..0900...01 with 13139 nines and
13139 zeros, or (5*10^13139-1)*10^13140+1 whose square is the concatenation of (5*10^13139-1)^2 and (10^13140-1)^2.

P has 26280 digits and has been certified prime with Pfgw (due to the
fact that a partial factorization of P-1 can be easily found).

Another way that allows to find large primes with the property required by Q1 is the following:
Fix a square, say 227^2 = 51529, and then search for a prime y
whose square is the concatenation of a square x^2
and 51529.
This can be accomplished by solving the diophantine equation
y^2 = 100000*x^2 + 51529
and then generating pairs of solutions (x,y) up to a certain limit and selecting those for which y is prime. For example, in this case I obtain P=32494189635056477 and
P^2=10558723600390117819210349796 | 51529 =
102755649968214^2 | 227^2.

Using this technique I found some primes with hundreds of digits.

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