Sun · The 1–3–5 conjecture

Every nonnegative integer n can be written as x²+y²+z²+w² with nonnegative integers x,y,z,w such that x+3y+5z is itself a square.

four squaresrepresentationsquadratic forms

The problem

Every nonnegative integer n can be written as x²+y²+z²+w² with nonnegative integers x,y,z,w such that x+3y+5z is itself a square.

Open since2016exact formulation date

Last checked2026.07.27Catalog verification

Offer 01$1,350Zhi-Wei Sun

Personal offer

The official slides offer US$1,350 for the first solution of the conjecture.