Sets of 4 Integers a, b, c, d for which Every Integer is in form ax^2 + by^2 + cz^2 + du^2

Theorem

There are exactly $55$ sets of $4$ integers $\left\{ {a, b, c, d}\right\}$ such that all integers can be written in the form:

$n = a x^2 + b y^2 + c z^2 + d w^2$

for integer $x, y, z, w$.

Proof

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Sources

• 1927: G.H. Hardy, P.V. Seshu Aiyar and B.M. Wilson: Collected Papers of Srinivasa Ramanujan
• 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $55$
• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $55$