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PreprintCC BYZenodo (CERN European Organization for Nuclear Research)

Prescribed digit sums in residue classes and a conjecture of Murthy (OEIS A077754)

  • Frédéric Gaudin1
  1. 1University of Caen
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Abstract

We prove an equidistribution theorem for numbers with all-nonzero decimal digits and prescribed digit sum and residue class. As a corollary, we resolve Murthy's conjecture (OEIS A077754, 2002): an integer n admits a multiple with at least two nonzero digits and digit sum n iff n is not in {1, 2, 5, 11} ∪ 10ℕ. The corollary is fully machine-verified in Lean 4 with mathlib (theorem A077754, zero sorry). The deposit includes the paper (paper.pdf), the complete Lean formalization (A077754.lean, CornerTable.lean), an independent Python verifier (~90 lines, stdlib) re-deriving all finite certificates, and the pinned build files (lakefile.toml, lake-manifest.json, lean-toolchain) for long-term reproducibility.

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Cite

BibTeXRIS
BibTeX
@misc{gaudin2026prescribed,
  title = {{Prescribed digit sums in residue classes and a conjecture of Murthy (OEIS A077754)}},
  author = {Frédéric Gaudin},
  howpublished = {Zenodo (CERN European Organization for Nuclear Research)},
  year = {2026},
  month = nov,
  doi = {10.5281/zenodo.23879087},
  url = {https://doi.org/10.5281/zenodo.23879087},
  note = {CC BY},
}
RIS
TY  - UNPB
TI  - Prescribed digit sums in residue classes and a conjecture of Murthy (OEIS A077754)
AU  - Frédéric Gaudin
PY  - 2026
DA  - 2026/11/02
PB  - Zenodo (CERN European Organization for Nuclear Research)
DO  - 10.5281/zenodo.23879087
UR  - https://doi.org/10.5281/zenodo.23879087
AB  - We prove an equidistribution theorem for numbers with all-nonzero decimal digits and prescribed digit sum and residue class. As a corollary, we resolve Murthy's conjecture (OEIS A077754, 2002): an integer n admits a multiple with at least two nonzero digits and digit sum n iff n is not in {1, 2, 5, 11} ∪ 10ℕ. The corollary is fully machine-verified in Lean 4 with mathlib (theorem A077754, zero sorry). The deposit includes the paper (paper.pdf), the complete Lean formalization (A077754.lean, CornerTable.lean), an independent Python verifier (~90 lines, stdlib) re-deriving all finite certificates, and the pinned build files (lakefile.toml, lake-manifest.json, lean-toolchain) for long-term reproducibility.
ER  -