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Extended Euclid gives $73 = 3^{2} + 8^{2}$ from the root $27$ of $- 1$

Example

Starting from $27^{2} \equiv - 1 (m o d 73)$ , the Euclidean algorithm yields $73 = 3^{2} + 8^{2}$ without searching the full collision square.

Facts & Assumptions

Given: The congruence $27^{2} \equiv - 1 (m o d 73)$ .

[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

The Euclidean divisions $73 = 2 \cdot 27 + 19$ and $27 = 19 + 8$ give $8 = 27 - 19 = 3 \cdot 27 - 73$ .

algebra

2.1

Hence $8 \equiv 3 \cdot 27 (m o d 73)$ , and the given square-root congruence gives $8^{2} \equiv 9 \cdot 27^{2} \equiv - 9 (m o d 73)$ ; therefore $73 ∣ 3^{2} + 8^{2}$ .

step 1.1algebra

3.1

Direct evaluation gives $3^{2} + 8^{2} = 9 + 64 = 73$ , so $(3, 8)$ is a two-square representation of $73$ by [F1].

step 2.1F1algebra∎

Depends on

Representations and primitive representations as sums of two squares

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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

Extended Euclid gives 73=32+8273=32+82 from the root 2727 of −1−1