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A316833 Sums of four distinct odd squares.

%I A316833 #25 Jul 20 2018 22:29:17
%S A316833 84,116,140,156,164,180,196,204,212,228,236,244,252,260,276,284,300,
%T A316833 308,316,324,332,340,348,356,364,372,380,396,404,420,428,436,444,452,
%U A316833 460,468,476,484,492,500,508,516,524,532,540,548,556,564,572,580,588,596,604,612,620,628,636,644,652,660
%N A316833 Sums of four distinct odd squares.
%C A316833 Theorem (Conjectured by R. William Gosper, proved by M. D. Hirschhorn): Any sum of four distinct odd squares is the sum of four distinct even squares.
%C A316833 The proof uses the following identity:
%C A316833 (4a+1)^2+(4b+1)^2+(4c+1)^2+(4d+1)^2 = 4[ (a+b+c+d+1)^2 + (a-b-c+d)^2 + (a-b+c-d)^2 + (a+b-c-d)^2 ].
%C A316833 All terms == 4 (mod 8).  Are all numbers == 4 (mod 8) and > 412 members of the sequence? - _Robert Israel_, Jul 20 2018
%D A316833 R. William Gosper and Stephen K. Lucas, Postings to Math Fun Mailing List, July 19 2018
%D A316833 Michael D. Hirschhorn, The Power of q: A Personal Journey, Springer 2017. See Chapter 31.
%H A316833 Robert Israel, <a href="/A316833/b316833.txt">Table of n, a(n) for n = 1..10000</a>
%p A316833 N:= 1000: # to get all terms <= N
%p A316833 V:= Vector(N):
%p A316833 for a from 1 to floor(sqrt(N/4)) by 2 do
%p A316833   for b from a+2 to floor(sqrt((N-a^2)/3)) by 2 do
%p A316833     for c from b+2 to floor(sqrt((N-a^2-b^2)/2)) by 2 do
%p A316833       for d from c + 2  by 2 do
%p A316833         r:= a^2+b^2+c^2+d^2;
%p A316833         if r > N then break fi;
%p A316833         V[r]:= V[r]+1
%p A316833 od od od od:
%p A316833 select(t -> V[t]>=1, [$1..N]); # _Robert Israel_, Jul 20 2018
%Y A316833 A316834 lists the subsequence for which the representation is unique.
%Y A316833 Cf. A004433, A316835.
%K A316833 nonn
%O A316833 1,1
%A A316833 _N. J. A. Sloane_, Jul 19 2018