[seqfan] pentolysis: coming apart in five pieces

wouter meeussen wouter.meeussen at pandora.be
Sat Apr 17 23:39:55 CEST 2010


a minor surprise in extending (.. specialising?) Fermat's 4n+1 theorem, that
says :
a prime number of the form 4n+1 can be written uniquely as a^2+b^2.

Pentolysis:
IF a prime number of the form 4n+1 can be written as a sum of distinct odd
squares,
THEN only in a sum of 5 distinct odd squares,
or only in a sum of 9 distinct odd squares;
or both in a sum of as well 5 as 13 distinct odd squares,
but no such prime can be written as a sum of  5 as well as of 9 distinct odd
squares.
And never only as a sum of 13.

True?? The 'why' escapes me, but I'd like to know.

Wouter.
---------------

Details and examples:

The least 4n+1 prime that comes apart at all  is
277 = 1+25+49+81+121 say  sumofsquares{1,5,7,9,11},

further:
227={1,5,7,9,11}
349={1,3,7,11,13}
373={3,5,7,11,13}={1,3,5,7,17}
389={3,5,7,9,15}
397={1,5,9,11,13}
421={1,7,9,11,13}={1,5,7,11,15}
461={3,5,9,11,15}
509={3,5,9,13,15}
541={1,5,11,13,15}={1,7,9,11,17}={3,5,7,13,17}={1,3,7,11,19}
557={1,5,7,11,19}={1,3,5,9,21}
613={5,7,9,13,17}={3,5,11,13,17}={1,7,9,11,19}={3,5,7,13,19}={1,3,5,7,23}
...

Since an odd square always equals 1 mod 4, you either need 1 (ahum..) or 5
or 9 or 1+4n of them.
What is the least 4n+1-prime that comes apart into 9 different odd squares?
It's 1249=1+9+25+49+121+169+225+289+361 say
sumofsquares{1,3,5,7,11,13,15,17,19}

The least 4n+1 prime that comes apart into 13 distinct odd squares is
3389 = 1+9+25+49+81+121+169+225+289+361+441+529+1089 = 289+441+729+841+1089
= ...
and the rule seems that whatever comes apart in 13 ways, also comes apart in
5 ways.

_____ the usual toxics beyond this point should not be too irritant_____

p1[k_Integer]:=p1[k]=Part[Select[Prime[Range[4k]],(Mod[#,4]==1)&],k];
Table[Expand at Coefficient[Series[Product[(1+q[(2k-1)^2]x^(2k-1)^2),
{k,Floor[1+Sqrt[p1[w]]/2]}],{x,0,p1[w]}],x^p1[w]],{w,234,234}]
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