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)

(See also Riemann Hypothesis on page 918.)