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feat: add repunit theorem helpers #1892
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nickzerjeski
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,79 @@ | ||
| /** | ||
| * Repunit theorem helpers. | ||
| * | ||
| * A repunit of length n is: | ||
| * R_n = (10^n - 1) / 9 | ||
| * | ||
| * For a prime p (p != 2, 3, 5), p divides R_n iff ord_p(10) divides n. | ||
| * Reference: https://en.wikipedia.org/wiki/Repunit | ||
| */ | ||
|
|
||
| const gcd = (a, b) => { | ||
| let x = BigInt(a) | ||
| let y = BigInt(b) | ||
| while (y !== 0n) { | ||
| ;[x, y] = [y, x % y] | ||
| } | ||
| return x < 0n ? -x : x | ||
| } | ||
|
|
||
| const modPow = (base, exp, mod) => { | ||
| let result = 1n | ||
| let b = BigInt(base) % BigInt(mod) | ||
| let e = BigInt(exp) | ||
| const m = BigInt(mod) | ||
|
|
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| while (e > 0n) { | ||
| if (e & 1n) result = (result * b) % m | ||
| b = (b * b) % m | ||
| e >>= 1n | ||
| } | ||
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| return result | ||
| } | ||
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| const multiplicativeOrder10 = (prime) => { | ||
| const p = BigInt(prime) | ||
| if (p <= 1n) throw new RangeError('prime must be > 1') | ||
| if (gcd(10n, p) !== 1n) throw new RangeError('10 and prime must be coprime') | ||
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| // For prime p, ord_p(10) divides p-1. | ||
| const upper = p - 1n | ||
| for (let k = 1n; k <= upper; k++) { | ||
| if (upper % k === 0n && modPow(10n, k, p) === 1n) { | ||
| return k | ||
| } | ||
| } | ||
|
nickzerjeski marked this conversation as resolved.
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| throw new Error('multiplicative order not found') | ||
| } | ||
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| const repunitMod = (length, mod) => { | ||
| if (!Number.isInteger(length) || length < 1) { | ||
| throw new RangeError('length must be a positive integer') | ||
| } | ||
| const m = BigInt(mod) | ||
| if (m <= 0n) throw new RangeError('mod must be > 0') | ||
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| let remainder = 0n | ||
| for (let i = 0; i < length; i++) { | ||
| remainder = (remainder * 10n + 1n) % m | ||
| } | ||
| return remainder | ||
| } | ||
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| const isRepunitDivisibleByPrime = (length, prime) => { | ||
| if (!Number.isInteger(length) || length < 1) { | ||
| throw new RangeError('length must be a positive integer') | ||
| } | ||
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| const p = BigInt(prime) | ||
| if (p === 2n || p === 5n) return false | ||
| if (p === 3n) return BigInt(length) % 3n === 0n | ||
| if (gcd(10n, p) !== 1n) return false | ||
|
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| const order = multiplicativeOrder10(p) | ||
| return BigInt(length) % order === 0n | ||
|
nickzerjeski marked this conversation as resolved.
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| } | ||
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| export { multiplicativeOrder10, repunitMod, isRepunitDivisibleByPrime } | ||
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,50 @@ | ||
| import { | ||
| isRepunitDivisibleByPrime, | ||
| multiplicativeOrder10, | ||
| repunitMod | ||
| } from '../RepunitTheorem' | ||
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| describe('RepunitTheorem', () => { | ||
| it('computes multiplicative order examples', () => { | ||
| expect(multiplicativeOrder10(11n)).toBe(2n) | ||
| expect(multiplicativeOrder10(37n)).toBe(3n) | ||
| expect(multiplicativeOrder10(7n)).toBe(6n) | ||
| }) | ||
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| it('checks repunit divisibility using the theorem', () => { | ||
| // 111111 is divisible by 3, 7, 11, 13, 37 | ||
| expect(isRepunitDivisibleByPrime(6, 3n)).toBe(true) | ||
| expect(isRepunitDivisibleByPrime(6, 7n)).toBe(true) | ||
| expect(isRepunitDivisibleByPrime(6, 11n)).toBe(true) | ||
| expect(isRepunitDivisibleByPrime(6, 13n)).toBe(true) | ||
| expect(isRepunitDivisibleByPrime(6, 37n)).toBe(true) | ||
| }) | ||
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| it('returns false when divisibility condition does not hold', () => { | ||
| expect(isRepunitDivisibleByPrime(1, 3n)).toBe(false) | ||
| expect(isRepunitDivisibleByPrime(3, 3n)).toBe(true) | ||
| expect(isRepunitDivisibleByPrime(6, 19n)).toBe(false) | ||
| expect(isRepunitDivisibleByPrime(9, 2n)).toBe(false) | ||
| expect(isRepunitDivisibleByPrime(9, 5n)).toBe(false) | ||
| }) | ||
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nickzerjeski marked this conversation as resolved.
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| it('computes repunit modulo without building huge integers', () => { | ||
| expect(repunitMod(6, 37n)).toBe(0n) | ||
| expect(repunitMod(6, 11n)).toBe(0n) | ||
| expect(repunitMod(7, 13n)).toBe(1n) | ||
| }) | ||
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| it('validates multiplicative order input', () => { | ||
| expect(() => multiplicativeOrder10(1n)).toThrow(RangeError) | ||
| expect(() => multiplicativeOrder10(10n)).toThrow(RangeError) | ||
| }) | ||
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| it('validates repunitMod input', () => { | ||
| expect(() => repunitMod(0, 7n)).toThrow(RangeError) | ||
| expect(() => repunitMod(5, 0n)).toThrow(RangeError) | ||
| }) | ||
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| it('validates repunit divisibility input length', () => { | ||
| expect(() => isRepunitDivisibleByPrime(0, 7n)).toThrow(RangeError) | ||
| }) | ||
| }) | ||
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The file header comment says the theorem holds for primes p != 2,5, but it also needs p != 3 (equivalently gcd(p,9)=1). As written, it documents an incorrect condition and matches the buggy behavior for p=3.