A Sharp Bound for a Shifted GCD Sum over Divisors

For every integer n >= 2, defineA(n) = sum_{d|n} gcd(d-1, n/d-1).We prove the sharp inequality A(n) <= 9n/4. Equality holds if and onlyif n = 36. The argument is elementary apart from an exact finiteverification for 2 <= n <= 1,340,963; complete source code for thatverification is supplied. We also record a reformulation as a restrictedshifted gcd sum modulo n-1.

Authors

Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-04
DOI
https://doi.org/10.5281/zenodo.22313899
Primary Topic
Coding theory and cryptography
Type
preprint
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preprint

A Sharp Bound for a Shifted GCD Sum over Divisors

Krystal Hirsch
Zenodo (CERN European Organization for Nuclear Research)
Coding theory and cryptography
preprint

A Sharp Bound for a Shifted GCD Sum over Divisors

Krystal Hirsch
preprint en

Abstract

For every integer n >= 2, defineA(n) = sum_{d|n} gcd(d-1, n/d-1).We prove the sharp inequality A(n) <= 9n/4. Equality holds if and onlyif n = 36. The argument is elementary apart from an exact finiteverification for 2 <= n <= 1,340,963; complete source code for thatverification is supplied. We also record a reformulation as a restrictedshifted gcd sum modulo n-1.

Zenodo (CERN European Organization for Nuclear Research)
Reduced inequalities
Coding theory and cryptography
DOIOpen Access PDFOpenAlex
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