# m=p^2+q^2, (p<q primes): number of presentations

zak seidov zakseidov at yahoo.com
Tue Feb 26 20:21:17 CET 2008

```Dear seqfans,

first of all,
many thanks to you those responding
(and sorry to much more those annoying..)
to my question_posts.

Here is my last one.
Can anyone plz check and extend this?
thanks, zak

A1:
Smallest positive number expressible as sum of squares
of two distinct primes p<q exactly in n ways:
1, 13, 410, 2210, 10370, 202130, 229970, 197210, 81770
0 way: 1
1 way: 13=2^2+3^2,
2 ways: 410=7^2+19^2=11^2+17^2,
3 ways: 2210=19^2+43^2=23^2+41^2=29^2+37^2,
4 ways:
10370=13^2+101^2=31^2+97^2=59^2+83^2=71^2+73^2,
5 ways:
202130=23^2+449^2=97^2+439^2=163^2+419^2=211^2+397^2=251^2+373^2,

6 ways:
229970=23^2+479^2=109^2+467^2=193^2+439^2=263^2+401^2=269^2+397^2=331^2+347^2,
7 ways:
197210=31^2+443^2=67^2+439^2=107^2+431^2=173^2+409^2=199^2+397^2=241^2+373^2=311^2+317^2,
8 ways:
81770=41^2+283^2=53^2+281^2=71^2+277^2=97^2+269^2=137^2+251^2=157^2+239^2=179^2+223^2=193^2+211^2,
9 ways: ?
10 ways: ?
...........
...........

BTW for those interested:
for m's up to 10^6,
there are only 8732 m's
expressible as p^2+q^2 (p<q primes)
at least in 1 way,
right?

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