Chapter 12: The Principle of Computational Equivalence

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; 4th 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 n2+1? (Quadratic primes; 1840s?) (See page 1162.)

• Are there infinitely many primes of the form 22n+1? (Fermat primes; 1844)

• Are there no solutions to xm-yn==1 other than 32-23==1? (Catalan's Conjecture; 1844)

• Can every integer not of the form 9n±4 be written as a3±b3±c3? (See note above.)

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

(See also Riemann Hypothesis on page 918.)

From Stephen Wolfram: A New Kind of Science [citation]