slug: oeis-a290322-cyclotomic-five-residue-sum bibkey: oeis2024a290322 doi: null url: https://oeis.org/A290322 triage: theorem motivation_gids:

• D5/S3/Arith/CyclotomicFiveResidueSum

The A290322 admissible residue sum conjecture

Problem

For a modulus at least two, call a residue below it admissible when it is coprime to the modulus and the value of u^4 + u^3 + u^2 + u + 1 at it is coprime to the modulus as well. Robert Israel conjectured on January 23, 2024 that the sum of the admissible residues is not divisible by the modulus whenever five divides the modulus.

Motivation

This is a first-tier recent OEIS conjecture. Direct inspection on September 9, 2026 found the statement still labelled Conjecture. The companion sequence A290309 counts the same residues and is marked easy; the weighted first moment is a different function and the counting results do not settle it.

Gap

The searched public indexes returned no proof of this statement and no earlier weighted-sum formula from which it follows. The polynomial-totient literature reached through Csizmazia and Toth weights by a greatest common divisor rather than by the residue, so its Euler products do not apply.

Route

Write the modulus as a power of five times a factor prime to five, and let the exponent of five be at least one.

The Chinese remainder theorem splits the admissible set as a product, and each admissible residue of the five-part appears once for every admissible residue of the other part. So the sum is congruent, modulo the power of five, to the count on the prime-to-five part times the sum on the five-part.

Modulo five the cyclotomic value vanishes exactly at one, so the admissible residues are exactly those congruent to two, three or four. Their representatives modulo the power of five are those three residues shifted by multiples of five, which gives a closed form for the sum by summing an arithmetic progression. That closed form is congruent to the negative of the next lower power of five.

For the remaining factor, over a prime field the product of the cyclotomic value with one less than the variable is the fifth power minus one, and away from characteristic five the value one is not a root, so the excluded units are exactly the nontrivial fifth roots of unity. The unit group is cyclic, so their number is the greatest common divisor of five with one less than the prime. Either way the resulting count is prime to five, and multiplicativity extends this to the whole factor.

Combining the three, the five-adic valuation of the sum is exactly one less than that of the modulus, hence strictly smaller, so the modulus cannot divide the sum.

Falsifier

A modulus divisible by five whose admissible residues sum to a multiple of it would contradict the theorem about the defined sum.

Evidence

• Module: D5/S3/Arith/CyclotomicFiveResidueSum.lean.
• Main theorem: residue_sum_ne_zero.
• Supporting public results: phi5_mod_five_eq_zero_iff, sum_goodUnits_five_pow, residue_sum_five_pow_ne_zero, residueCount_mul, sum_goodUnits_mul_cast, residueCount_pow, residueCount_prime, residueCount_not_dvd_five.
• The module reuses D5/S3/Arith/ChineseRemainder rather than rebuilding the splitting.
• There is no finite cutoff in any public statement.

The caller computed the sums directly before dispatching an implementation seat, and checked eight assertions rather than accepting the reduction it was given. The conjecture itself held for all forty moduli divisible by five up to two hundred, with no violation; for moduli not divisible by five the sum was a multiple of the modulus in seventy-eight cases below one hundred and twenty, which shows the divisibility hypothesis is doing real work rather than being decorative. The cyclotomic value was congruent to zero modulo five exactly at one. The admissible set for a power of five was exactly the residues congruent to two, three or four, checked for the first three powers. The closed form for the sum over a power of five was an exact equality for the first four powers, and its congruence to the negative of the next lower power held for the first six. The count formula for prime powers matched on thirteen primes for the first two exponents with no mismatch, and no modulus prime to five below three hundred had a count divisible by five.

Triage

theorem. The conjecture is proved for every modulus divisible by five. The module additionally records the exact five-adic valuation, which is stronger than the nonvanishing that was asked for. Nothing is asserted about moduli not divisible by five, where the sum frequently is a multiple of the modulus.

ASSUMED-UNVERIFIED

First-publication priority is not established. The searches do not exclude private or unindexed proofs. The identification with the OEIS entry is documentary; the kernel verifies the explicitly defined sum.