Re: Integers that are the sum of 2 nonzero squares in exactly 2 ways

• To: mathgroup at smc.vnet.net
• Subject: [mg125748] Re: Integers that are the sum of 2 nonzero squares in exactly 2 ways
• From: danl at wolfram.com
• Date: Sat, 31 Mar 2012 03:43:19 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <jkungd\$asj\$1@smc.vnet.net>

```On Wednesday, March 28, 2012 5:00:45 AM UTC-5, Cisco Lane wrote:
> I've been looking at integers that are the sum of 2 nonzero squares in exactly 2 ways. The smallest example is 50 = 5^2+5^2=7^2+1^2. The first few terms are 50, 65, 85, 125, 130, 145, ....  This is given in OEIS as https://oeis.org/A025285
>
> If I plot the pairs {1,50},{2,65},{3,85},... I get a more or less straight line with a slope of about 8.85... In other words, eventually, about one in 8.85 integers qualify.
>
> I wonder if there is a theoretical value for this approximate number of 8.85...?

I don't know if this answers anything, but the factorization pattern of such numbers appears to be as follows. All primes of the form 4n+3 occur to even powers. There are one, two or three other factors. They can take the form

2^n*p*q where p and q are distinct primes of the form 4n+1, n an arbitrary nonnegative integer

2^(2*n+1)*p^2 with n, p as above

2^(2*n)*p^(2*k) for n as above and k satisfying some relation I cannot quite figure out other than it has to be at least 2. There may also be further restrictions on n that depend on k.

Daniel Lichtblau
Wolfram Research

```

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