FAMOUS CONJECTURES

Goldbach's conjecture says that every even number (except 2) is the sum of two primes. It has been checked up to 4*10^11 by Sinisalo in 1993, up to 10^14 by Jean-Marc Deshouillers, Yannick Saouter and Herman te Riele in 1997 (paper submitted to ANTS3). Olivier Ramaré proved in 1995 that every even integer is the sum of at most 6 primes.

Sums of cubes. Together with Olivier Ramaré and Francois Bertault, we have proved that every integer between 1,290,741 and 15000^3 = 3,375,000,000,000 is a sum of five nonnegative cubes by exhaustive search [paper]. Using some results on sums of 4 cubes, Jean-Marc Deshouillers, Francois Hennecart and Bernard Landreau have improved this result to 10^16 (for sums of 5 cubes). Jean-Marc Deshouillers, Francois Hennecart and Bernard Landreau conjecture that every integer greater than 7,373,170,279,850 is a sum of 4 cubes.