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Thue's collision argument gives $73 = 3^{2} + 8^{2}$

Example

The prime $73$ has $27^{2} \equiv - 1 (m o d 73)$ . Thue's collision construction with the square $0 \leq i, j \leq 8$ produces the representation

$73 = 3^{2} + 8^{2} .$

Facts & Assumptions

Given: The prime $p = 73$ and the residue $a = 27$ .

[L1]

If $p$ is prime and $p ∤ a$ , then there are nonzero integers $r, s$ with $∣ r ∣, ∣ s ∣ < \sqrt{p}$ and $r a \equiv s (m o d p)$ (Thue's lemma on small nonzero representatives).

[F1]

A representation of a nonnegative integer $n$ as a sum of two squares is an ordered pair $(x, y) \in Z^{2}$ such that $n = x^{2} + y^{2}$ (Representations and primitive representations as sums of two squares).

Verification

technique · direct

1.1

One has $27^{2} = 729 \equiv - 1 (m o d 73)$ , and $8^{2} = 64 < 73 < 81 = 9^{2}$ , so the maximal integer $h$ with $h^{2} < 73$ is $8$ .

algebra

2.1

In the map $(i, j) \mapsto 27 i - j (m o d 73)$ on $0 \leq i, j \leq 8$ , the distinct pairs $(3, 8)$ and $(0, 0)$ collide because $3 \cdot 27 - 8 = 73$ . Their coordinate differences are $(3, 8)$ , exactly as in [L1].

step 1.1L1algebra

3.1

The bounds give $0 < 3^{2} + 8^{2} < 2 \cdot 73$ , and step 2.1 together with $27^{2} \equiv - 1$ makes the sum divisible by $73$ . Directly, $3^{2} + 8^{2} = 9 + 64 = 73$ , so $(3, 8)$ is a representation in the sense of [F1].

step 1.1step 2.1L1F1algebra∎

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Sources

P. Hackman, Elementary Number Theory, Chapter E, Example E.I.4 (standard reference, not scraped)

Thue's collision argument gives 73=32+8273=32+82