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Topic: Sum of two squares problem
Replies: 3   Last Post: Jul 27, 2010 7:44 PM

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 sales@kt-algorithms.com Posts: 78 Registered: 1/29/05
Sum of two squares problem
Posted: Jul 27, 2010 7:36 AM

For the sum of two squares function (1):

r2(c) = number of integer pairs (x,y) for which x^2 + y^2 = c

consider the special arguments c' defined by:

r2(c') > sup{r2(d): d < c'}

A table of c' and r2(c') starts like this:

-------------------------------------------------------------------
c' r2(c') Factors of r2(c')
-------------------------------------------------------------------
1 4 1
5 8 5
25 12 5^2
65 16 5 13
325 24 5^2 13
1105 32 5 13 17
4225 36 5^2 13^2
-------------------------------------------------------------------
5525 48 5 (5 13 17)
27625 64 5^2 (5 13 17)
71825 72 5 13 (5 13 17)
138125 80 5^3 (5 13 17)
-------------------------------------------------------------------
160225 96 5 (5 13 17 29)
801125 128 5^2 (5 13 17 29)
2082925 144 5 13 (5 13 17 29)
4005625 160 5^3 (5 13 17 29)
-------------------------------------------------------------------
5928325 192 5 (5 13 17 29 37)
29641625 256 5^2 (5 13 17 29 37)
77068225 288 5 13 (5 13 17 29 37)
148208125 320 5^3 (5 13 17 29 37)
-------------------------------------------------------------------
243061325 384 5 (5 13 17 29 37 41) ?
1215306625 512 5^2 (5 13 17 29 37 41) ?
3159797225 576 5 13 (5 13 17 29 37 41) ?
6076533125 640 5^3 (5 13 17 29 37 41) ?
-------------------------------------------------------------------
12882250225 768 5 (5 13 17 29 37 41 53) ?
64411251125 1024 5^2 (5 13 17 29 37 41 53) ?
167469252925 1152 5 13 (5 13 17 29 37 41 53) ?
322056255625 1280 5^3 (5 13 17 29 37 41 53) ?
-------------------------------------------------------------------
785817263725 1536 5 (5 13 17 29 37 41 53 61) ?
3929086318625 2048 5^2 (5 13 17 29 37 41 53 61) ?
10215624428425 2304 5 13 (5 13 17 29 37 41 53 61) ?
19645431593125 2560 5^3 (5 13 17 29 37 41 53 61) ?
-------------------------------------------------------------------
57364660251925 3072 5 (5 13 17 29 37 41 53 61 73) ?
286823301259625 4096 5^2 (5 13 17 29 37 41 53 61 73) ?
745740583275025 4608 5 13 (5 13 17 29 37 41 53 61 73) ?
1434116506298125 5120 5^3 (5 13 17 29 37 41 53 61 73) ?
-------------------------------------------------------------------
5105454762421325 6144 5 (5 13 17 29 37 41 53 61 73 89) ?
25527273812106625 8192 5^2 (5 13 17 29 37 41 53 61 73 89) ?
66370911911477225 9216 5 13 (5 13 17 29 37 41 53 61 73 89) ?
127636369060533125 10240 5^3 (5 13 17 29 37 41 53 61 73 89) ?
-------------------------------------------------------------------

The entries annotated with '?' are tentative, as for the alleged c'
only r2(c') and r2(c'-1) were calculated. The preceding entries,
however, are based on exhaustive searches.

It seems like c' can be factored into primes congruent to 1 modulo 4,
reminding one of Fermat's theorem on sums of two squares and the
Brahmagupta?Fibonacci identity.

Any comments on the apparent regularity for c'>=5525: each time c'
doubles, the next prime congruent to 1 modulo 4 is added as a factor?

Best regards,
Knud Thomsen
Geologist, Denmark

References
1. Mathworld Sum of Squares Function
http://mathworld.wolfram.com/SumofSquaresFunction.html

Date Subject Author
7/27/10 sales@kt-algorithms.com
7/27/10 gerry@math.mq.edu.au
7/27/10 sales@kt-algorithms.com
7/27/10 Gerry Myerson