Classification of the positive integers not being the sum of four non-zero squares

page 140 in Conway's little book, $$ 1,3,5,9,11,17,29,41, \; 2 \cdot 4^m \; , \; 6 \cdot 4^m \; , \; 14 \cdot 4^m \; . $$ The proof is on the same page, with preparatory material in the previous few pages.

The first detail: any number $3 \pmod 8$ is the sum of three squares, meanwhile they must be odd squares, therefore nonzero. The square of any number that is divisible by $4$ becomes $0 \pmod 8.$ As a result, any number $6 \pmod 8$ is the sum of three squares, as $ (2A)^2 + B^2 + C^2,$ where $A,B,C$ must be odd squares, therefore nonzero.

10 June: Second detail: if $x^2 + y^2 + z^2 \equiv 0 \pmod 4,$ then $x,y,z$ are all even. This means that $12 \pmod{32}$ is the sum of three nonzero squares. Same for $24 \pmod{32}$

Some of my topograph answers, in order by question number. I got better with the diagrams as time went by. If you just look at these, not much will happen. If you draw some of your own examples, you will begin to understand.

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BOOKS:

http://www.maths.ed.ac.uk/~aar/papers/conwaysens.pdf (Conway)

http://www.springer.com/us/book/9780387955872 (Stillwell)

https://www.math.cornell.edu/~hatcher/TN/TNbook.pdf (Hatcher)

http://bookstore.ams.org/mbk-105/ (Weissman)

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ANSWERS:

http://math.stackexchange.com/questions/81917/another-quadratic-diophantine-equation-how-do-i-proceed/144794#144794

http://math.stackexchange.com/questions/228356/how-to-find-solutions-of-x2-3y2-2/228405#228405

http://math.stackexchange.com/questions/342284/generate-solutions-of-quadratic-diophantine-equation/345128#345128

http://math.stackexchange.com/questions/487051/why-cant-the-alpertron-solve-this-pell-like-equation/487063#487063

http://math.stackexchange.com/questions/512621/finding-all-solutions-of-the-pell-type-equation-x2-5y2-4/512649#512649

http://math.stackexchange.com/questions/680972/if-m-n-in-mathbb-z-2-satisfies-3m2m-4n2n-then-m-n-is-a-perfect-square/686351#686351

http://math.stackexchange.com/questions/739752/how-to-solve-binary-form-ax2bxycy2-m-for-integer-and-rational-x-y/739765#739765 :::: 69 55

http://math.stackexchange.com/questions/742181/find-all-integer-solutions-for-the-equation-5x2-y2-4/756972#756972

http://math.stackexchange.com/questions/822503/positive-integer-n-such-that-2n1-3n1-are-both-perfect-squares/822517#822517

http://math.stackexchange.com/questions/1078450/maps-of-primitive-vectors-and-conways-river-has-anyone-built-this-in-sage/1078979#1078979

http://math.stackexchange.com/questions/1091310/infinitely-many-systems-of-23-consecutive-integers/1093382#1093382

http://math.stackexchange.com/questions/1132187/solve-the-following-equation-for-x-and-y/1132347#1132347 <1,-1,-1>

http://math.stackexchange.com/questions/1132799/finding-integers-of-the-form-3x2-xy-5y2-where-x-and-y-are-integers

http://math.stackexchange.com/questions/1221178/small-integral-representation-as-x2-2y2-in-pells-equation/1221280#1221280

http://math.stackexchange.com/questions/1404023/solving-the-equation-x2-7y2-3-over-integers/1404126#1404126

http://math.stackexchange.com/questions/1599211/solutions-to-diophantine-equations/1600010#1600010

http://math.stackexchange.com/questions/1667323/how-to-prove-that-the-roots-of-this-equation-are-integers/1667380#1667380

http://math.stackexchange.com/questions/1719280/does-the-pell-like-equation-x2-dy2-k-have-a-simple-recursion-like-x2-dy2

http://math.stackexchange.com/questions/1737385/if-d1-is-a-squarefree-integer-show-that-x2-dy2-c-gives-some-bounds-i/1737824#1737824 "seeds"

http://math.stackexchange.com/questions/1772594/find-all-natural-numbers-n-such-that-21n2-20-is-a-perfect-square/1773319#1773319

Is there a simple proof that if $(b-a)(b+a) = ab - 1$, then $a, b$ must be Fibonacci numbers? 1,1,-1; 1,11

To find all integral solutions of $3x^2 - 4y^2 = 11$