Notes

Section 9: Implications for Mathematics and Its Foundations

Unsolved problems [in number theory]

Problems in number theory that are simple to state (say in the notation of Peano arithmetic) but that so far remain unsolved include:

• Is there any odd number equal to the sum of its divisors? (Odd perfect number; 4^th century BC) (See page 911.)

• Are there infinitely many primes that differ by 2? (Twin Prime Conjecture; 1700s?) (See page 909.)

• Is there a cuboid in which all edges and all diagonals are of integer length? (Perfect cuboid; 1719)

• Is there any even number which is not the sum of two primes? (Goldbach's Conjecture; 1742) (See page 135.)

• Are there infinitely many primes of the form n^2+1? (Quadratic primes; 1840s?) (See page 1162.)

• Are there infinitely many primes of the form 2^2^n+1? (Fermat primes; 1844)

• Are there no solutions to x^m-y^n==1 other than 3^2-2^3==1? (Catalan's Conjecture; 1844)

• Can every integer not of the form 9n±4 be written as a^3±b^3±c^3? (See note above.)

• How few n^th powers need be added to get any given integer? (Waring's Problem; 1770)