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

  • To: mathgroup at smc.vnet.net
  • Subject: [mg125702] Re: Integers that are the sum of 2 nonzero squares in exactly 2 ways
  • From: Dana DeLouis <dana01 at me.com>
  • Date: Thu, 29 Mar 2012 03:02:03 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com

> the sum of 2 nonzero squares in exactly 2 ways. The smallest example is 50

Hi.  The number 50 actually has 12 if you include negative numbers.  :>o

SquaresR[2,50]
12

Plus@@({-5,+5}^2)
50
Plus@@({-7,-1}^2)
50
etc...

> I get a more or less straight line with a slope of about 8.85.

Not sure, but the first 3 seems to have a slope of about 17.

t=Table[{n,PowersRepresentations[n,2,2]},{n,86}];
tt=First/@Select[t,Length[Last[#]]==2 &] //Rest
{50,65,85}

FindFit[tt,a *x +b,{a,b},x] //Expand
{a->17.5,b->31.6667}

If we search the first 100,000 numbers, then the slope drops down to 11.8
Just curious... how many numbers did you go out?

t=Table[{n,PowersRepresentations[n,2,2]},{n,100000}];
tt=First/@Select[t,Length[Last[#]]==2 &] //Rest;

FindFit[tt,a *x +b,{a,b},x] //Expand
{a->11.8312,b->102.192}

Using Differences returns about the same number as slope:

tt//Differences//Mean//N
11.8238

= = = = = = = = = =
HTH :>)
Dana DeLouis
Mac & Math 8
= = = = = = = = = =



On Mar 28, 6:00 am, Cisco Lane <travl... at yahoo.com> 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 ashttps://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...?





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