Quanta Magazine · Science

Mathematicians Solve 55-Year-Old Rearrangement Conjecture

A 55-year-old conjecture by Ronald Graham, asking if integers can be rearranged so all partial sums are unique, has been solved. The solution involved multiple papers and leveraged randomness.

Graham's conjecture concerned numbers in modular arithmetic. For decades, the case of cyclically repeating numbers remained open, though a valid rearrangement was suspected.

The problem was solved across four papers, with the last in February 2026. Müyesser and Pokrovskiy tackled large sets using random orderings and fixes for zero-sum sequences.

Kravitz and Bedert solved the conjecture for very small sets in September 2024. Their work focused on tiny sets relative to the modulus 'p'.

A collaboration extended solutions to larger sets in August 2025, but a gap for medium sets persisted.

Sauermann and Pham closed this gap in February 2026 using anti-concentration and Fourier analysis, proving problematic arrangements were unlikely.

The proofs confirm Graham's conjecture for all set sizes with very large 'p'. Resolving it for all 'p' remains a technicality.

The solution confirms that even in constrained number systems, flexible structures and unique patterns can always be achieved.

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