TITLE: Classification of the positive integers not being the sum of four non-zero squares
QUESTION [6 upvotes]: It is well known that every positive integer is the sum of at most four perfect squares (including $1$).

But which positive integers are not the sum of four non-zero perfect squares ($1$ is still allowed as a perfect square) ?

I showed that the numbers $2^k$ , $2^k\cdot 3$ and $2^k\cdot 7$ with odd positive integer $k$ have this property. I checked the numbers upto $10^4$ and above $41$, no examples , other than those of the mentioned forms , occured. So my question is whether additional positive integers with the desired property exist.

REPLY [3 votes]: 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.
================
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)
==============
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 $3x^2 - 4y^2 = 11$