The statement of the result was taken from the paper cited below by {{AuthorRef|Paul T. Bateman}} and others: :''The largest odd integer not expressible as a sum of $4$ distinct non-zero squares with greatest common divisor $1$ is $157$.'' By a brute force exercise, we assemble all the sets of $4$ distinct non-zero integers and calculate the sum of their squares such that those sums are less than $157$. It is noted in passing that all such sets have a greatest common divisor of $1$. Thus we have: {{begin-eqn}} {{eqn | l = 39       | r = 1 + 4 + 9 + 25       | rr= = 1^2 + 2^2 + 3^2 + 5^2       | c =  }} {{eqn | l = 51       | r = 1 + 9 + 16 + 25       | rr= = 1^2 + 3^2 + 4^2 + 5^2       | c =  }} {{eqn | l = 57       | r = 1 + 4 + 16 + 36       | rr= = 1^2 + 2^2 + 4^2 + 6^2       | c =  }} {{eqn | l = 63       | r = 1 + 4 + 9 + 49       | rr= = 1^2 + 2^2 + 3^2 + 7^2       | c =  }} {{eqn | l = 65       | r = 4 + 9 + 16 + 36       | rr= = 2^2 + 3^2 + 4^2 + 6^2       | c =  }} {{eqn | l = 71       | r = 1 + 9 + 25 + 36       | rr= = 1^2 + 3^2 + 5^2 + 6^2       | c =  }} {{eqn | l = 75       | r = 1 + 9 + 16 + 49       | rr= = 1^2 + 3^2 + 4^2 + 7^2       | c =  }} {{eqn | l = 79       | r = 1 + 2 + 25 + 49       | rr= = 1^2 + 2^2 + 5^2 + 7^2       | c =  }} {{eqn | l = 81       | r = 4 + 16 + 25 + 36       | rr= = 2^2 + 4^2 + 5^2 + 6^2       | c =  }} {{eqn | l = 85       | r = 1 + 4 + 16 + 64       | rr= = 1^2 + 2^2 + 4^2 + 8^2       | c =  }} {{eqn | l = 87       | r = 4 + 9 + 25 + 49       | rr= = 2^2 + 3^2 + 5^2 + 7^2       | c =  }} {{eqn | l = 91       | r = 1 + 16 + 25 + 49       | rr= = 1^2 + 4^2 + 5^2 + 7^2       | c =  }} {{eqn | l = 93       | r = 4 + 9 + 16 + 64       | rr= = 2^2 + 3^2 + 4^2 + 8^2       | c =  }} {{eqn | l = 95       | r = 1 + 9 + 36 + 49       | rr= = 1^2 + 3^2 + 6^2 + 7^2       | c =  }} {{eqn | l = 99       | r = 1 + 9 + 25 + 64       | rr= = 1^2 + 3^2 + 5^2 + 8^2       | c =  }} {{eqn | r = 9 + 16 + 25 + 49       | rr= = 3^2 + 4^2 + 5^2 + 7^2       | c =  }} {{eqn | l = 105       | r = 4 + 16 + 36 + 47       | rr= = 2^2 + 4^2 + 6^2 + 7^2       | c =  }} {{eqn | l = 107       | r = 1 + 9 + 16 + 81       | rr= = 1^2 + 3^2 + 4^2 + 9^2       | c =  }} {{eqn | l = 109       | r = 4 + 16 + 25 + 64       | rr= = 2^2 + 4^2 + 5^2 + 8^2       | c =  }} {{eqn | l = 111       | r = 1 + 25 + 36 + 49       | rr= = 1^2 + 5^2 + 6^2 + 7^2       | c =  }} {{eqn | r = 1 + 4 + 25 + 81       | rr= = 1^2 + 2^2 + 5^2 + 9^2       | c =  }} {{eqn | l = 113       | r = 4 + 9 + 36 + 64       | rr= = 2^2 + 3^2 + 6^2 + 8^2       | c =  }} {{eqn | l = 117       | r = 1 + 16 + 36 + 64       | rr= = 1^2 + 4^2 + 6^2 + 8^2       | c =  }} {{eqn | l = 119       | r = 4 + 9 + 25 + 81       | rr= = 2^2 + 3^2 + 5^2 + 9^2       | c =  }} {{eqn | r = 9 + 25 + 36 + 49       | rr= = 3^2 + 5^2 + 6^2 + 7^2       | c =  }} {{eqn | l = 121       | r = 1 + 4 + 16 + 100       | rr= = 1^2 + 2^2 + 4^2 + 10^2       | c =  }} {{eqn | l = 123       | r = 1 + 9 + 49 + 64       | rr= = 1^2 + 3^2 + 7^2 + 8^2       | c =  }} {{eqn | r = 1 + 16 + 25 + 81       | rr= = 1^2 + 4^2 + 5^2 + 9^2       | c =  }} {{eqn | l = 125       | r = 9 + 16 + 36 + 64       | rr= = 3^2 + 4^2 + 6^2 + 8^2       | c =  }} {{eqn | l = 127       | r = 1 + 9 + 36 + 81       | rr= = 1^2 + 3^2 + 6^2 + 9^2       | c =  }} {{eqn | l = 129       | r = 4 + 9 + 16 + 100       | rr= = 2^2 + 3^2 + 4^2 + 10^2       | c =  }} {{eqn | r = 4 + 25 + 36 + 64       | rr= = 2^2 + 5^2 + 6^2 + 8^2       | c =  }} {{eqn | l = 131       | r = 9 + 16 + 25 + 81       | rr= = 3^2 + 4^2 + 5^2 + 9^2       | c =  }} {{eqn | l = 133       | r = 4 + 16 + 49 + 64       | rr= = 2^2 + 4^2 + 7^2 + 8^2       | c =  }} {{eqn | l = 135       | r = 1 + 4 + 9 + 121       | rr= = 1^2 + 2^2 + 3^2 + 11^2       | c =  }} {{eqn | r = 1 + 4 + 49 + 81       | rr= = 1^2 + 2^2 + 7^2 + 9^2       | c =  }} {{eqn | r = 1 + 9 + 25 + 100       | rr= = 1^2 + 3^2 + 5^2 + 10^2       | c =  }} {{eqn | l = 137       | r = 4 + 16 + 36 + 81       | rr= = 2^2 + 4^2 + 6^2 + 9^2       | c =  }} {{eqn | l = 139       | r = 1 + 25 + 49 + 64       | rr= = 1^2 + 5^2 + 7^2 + 8^2       | c =  }} {{eqn | l = 141       | r = 16 + 25 + 36 + 64       | rr= = 4^2 + 5^2 + 6^2 + 8^2       | c =  }} {{eqn | l = 143       | r = 1 + 25 + 36 + 81       | rr= = 1^2 + 5^2 + 6^2 + 9^2       | c =  }} {{eqn | r = 4 + 9 + 49 + 81       | rr= = 2^2 + 3^2 + 7^2 + 9^2       | c =  }} {{eqn | l = 145       | r = 4 + 16 + 25 + 100       | rr= = 2^2 + 4^2 + 5^2 + 10^2       | c =  }} {{eqn | l = 147       | r = 1 + 16 + 49 + 81       | rr= = 1^2 + 4^2 + 7^2 + 9^2       | c =  }} {{eqn | r = 1 + 9 + 16 + 121       | rr= = 1^2 + 3^2 + 4^2 + 11^2       | c =  }} {{eqn | r = 9 + 25 + 49 + 64       | rr= = 3^2 + 5^2 + 7^2 + 8^2       | c =  }} {{eqn | l = 149       | r = 1 + 4 + 25 + 121       | rr= = 1^2 + 2^2 + 5^2 + 11^2       | c =  }} {{eqn | r = 4 + 9 + 36 + 100       | rr= = 2^2 + 3^2 + 6^2 + 10^2       | c =  }} {{eqn | l = 151       | r = 9 + 25 + 36 + 81       | rr= = 3^2 + 5^2 + 6^2 + 9^2       | c =  }} {{eqn | l = 153       | r = 4 + 36 + 49 + 64       | rr= = 2^2 + 6^2 + 7^2 + 8^2       | c =  }} {{eqn | l = 155       | r = 9 + 16 + 49 + 81       | rr= = 3^2 + 4^2 + 7^2 + 9^2       | c =  }} {{end-eqn}} It remains to be demonstrated that the above sequence is indeed complete. The original article by {{AuthorRef|Franz Halter-Koch}} states: :{{SmallCaps|Satz}} $3$. ''Genau dann is $n \in N$ nicht Summe von vier verschiedenen positiven (im Falle $n \not \equiv 0 \bmod 8$ auch teilerfremden) Quadraten, wenn entwieder'' ::''$n = 4^h a \quad$ mit $h \ge 0$ und $a \in \set {1, 2, \ldots, 19, 22, 23, 25, 26, 27, 31, 33, 34, 37, 38, 42, 43, 47, 55, 58, 67, 73, 82, 97, 103}$'' :''oder'' ::''$n \in \set {21, 29, 35, 41, 45, 49, 53, 59, 61, 69, 77, 83, 89, 101, 115, 157}$ That is, in English: :''{{SmallCaps|Statement}} $3$. $n \in N$ is not the sum of four different positive squares (in the case where $n \not \equiv 0 \bmod 8$ also non-prime) if and only if: ::''$n = 4^h a \quad$ where $h \ge 0$ and $a \in \set {1, 2, \ldots, 19, 22, 23, 25, 26, 27, 31, 33, 34, 37, 38, 42, 43, 47, 55, 58, 67, 73, 82, 97, 103}$ :''or'' ::''$n \in \set {21, 29, 35, 41, 45, 49, 53, 59, 61, 69, 77, 83, 89, 101, 115, 157}$ {{ProofWanted}}