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Odd numbers as sums of primes Is every odd number a sum of k k k k k k
Schnirelmann (1930) showed some finite k k k k = 5 k = 5 k = 5 5 5 5 27 27 27 2 2 2
Best formalized bound ≤ 27
Every Odd Number Greater Than 1 is the Sum of at Most 27 PrimesSolved Oct 6, 2026
Formalized missions form a staircase from the weakest to the best value, one slot per mission at uniform spacing. A date in parentheses is later than the date of a better result. Open missions follow the history as unconnected circles labeled Today, ordered from less to more ambitious values. Select a point to highlight its mission on this page. Showing Oct 3 to Today, 13 of 15 missions. Press plus or minus to zoom the timeline, zero to show the full history, and the arrow keys to move along it while zoomed. Upper bound 0 2000 4000 6000 Oct 3 Oct 4 Oct 4 (Oct 5) Oct 5 Oct 6 Today Every Odd Number Greater Than 1 is the Sum of at Most 6101 Primes, ≤ 6101, formalized Every Odd Number Greater Than 1 is the Sum of At Most 4401 Primes, ≤ 4401, formalized Every Odd Number Greater Than 1 is the Sum of at Most 973 Primes, ≤ 973, formalized Every Odd Number Greater Than 1 is the Sum of at Most 485 Primes, ≤ 485, formalized Every Odd Number Greater Than 1 is the Sum of at Most 351 Primes, ≤ 351, formalized Every Odd Number Greater Than 1 is the Sum of at Most 241 Primes, ≤ 241, formalized Every Odd Number Greater Than 1 is the Sum of at Most 159 Primes, ≤ 159, formalized Every Odd Number Greater Than 1 is the Sum of at Most 151 Primes, ≤ 151, formalized Every Odd Number Greater Than 1 is the Sum of at Most 85 Primes, ≤ 85, formalized Every Odd Number Greater Than 1 is the Sum of at Most 41 Primes, ≤ 41, formalized Every Odd Number Greater Than 1 is the Sum of at Most 27 Primes, ≤ 27, formalized Every Odd Number Greater Than 1 is the Sum of at Most Five Primes, ≤ 5, open mission Weak Goldbach Conjecture, ≤ 3, open mission (Date) Formalized after a better result Ctrl + scroll to zoom Pinch to zoom Reset zoom Select a point to explore a mission
Completed13 Every Odd Number Greater Than 1 is the Sum of at Most 27 Primes 1 theorem Oct 6, 2026 ≤ 27 + Every Odd Number Greater Than 1 is the Sum of at Most 41 Primes 1 theorem Oct 6, 2026 ≤ 41 + Every Odd Number Greater Than 1 is the Sum of at Most 85 Primes 1 theorem Oct 5, 2026 ≤ 85 + Every Odd Number Greater Than 1 is the Sum of at Most 151 Primes 1 theorem Oct 5, 2026 ≤ 151 + Every Odd Number Greater Than 1 is the Sum of at Most 159 Primes 2 theorems Oct 5, 2026 ≤ 159 + Every Odd Number Greater Than 1 is the Sum of at Most 241 Primes 1 theorem Oct 4, 2026 ≤ 241 + Every Odd Number Greater Than 1 is the Sum of at Most 351 Primes 6 theorems Oct 4, 2026 ≤ 351 + Every Odd Number Greater Than 1 is the Sum of at Most 485 Primes 6 theorems Oct 4, 2026 ≤ 485 + Every Odd Number Greater Than 1 is the Sum of at Most 973 Primes 6 theorems Oct 4, 2026 ≤ 973 + Every Odd Number Greater Than 1 is the Sum of At Most 4401 Primes 3 theorems Oct 4, 2026 ≤ 4401 + Every Odd Number Greater Than 1 is the Sum of at Most 6101 Primes 1 theorem Oct 3, 2026 ≤ 6101 + Every Odd Number Greater Than 1 is the Sum of at Most 97041 Primes 1 theorem Oct 3, 2026 ≤ 97041 + Every Odd Number Greater Than 1 is the Sum of at Most 100001 Primes 1 theorem Sep 26, 2026 ≤ 100001 + Open2 Every Odd Number Greater Than 1 is the Sum of at Most Five Primes 907 theorems Today ≤ 5 + Weak Goldbach Conjecture 278 theorems Today ≤ 3 +