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The four square roots of $9$ modulo $128$

Example

The square roots of $9$ modulo $128$ are

$3, 61, 67, 125.$

Facts & Assumptions

Given: The unit target $9$ modulo $128 = 2^{7}$ .

[L1]

For $k \geq 3$ , an odd target is a square modulo $2^{k}$ exactly when it is $1$ modulo $8$ , and every soluble target has exactly four roots (Unit square criterion and root count modulo powers of two).

Verification

technique · direct

1.1

The representatives are distinct modulo $128$ , and $3^{2} = 9$ , $61^{2} = 29 \cdot 128 + 9$ , $67^{2} = 35 \cdot 128 + 9$ , and $125^{2} = 122 \cdot 128 + 9$ . Thus all four displayed classes are roots.

givenalgebra

2.1

Since $9 \equiv 1 (m o d 8)$ , [L1] says that the congruence is soluble and has exactly four root classes. The four distinct roots in step 1.1 are therefore complete.

step 1.1L1∎

Depends on

Unit square criterion and root count modulo powers of two

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 8 results over 5 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

The four square roots of 99 modulo 128128