prime number

An integer greater than 1 that is divisible only by 1 and itself.

• 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, ...more... , 9973, ...
• Primes between 1 and 10000, patterns
• Jean-Francois Colonna's artistic view of prime numbers , between 1 and 262144

• Euclid proved that there are an infinite number of primes.
• "proofs" that every odd number is prime .

• The Fundamental Theorem of Arithmetic : Every integer > 1 can be decomposed as the product of primes, p ₁ ^{a ₁} ... p _n ^{a _n} , and this decomposition is unique .

  2 , 3 , 4=2 ² , 5 , 6=2×3, 7 , 8=2 ³ , 9=3 ² , 10=2×5, 11 , 12=2 ² ×3, 13 , 14=2×7, 15=3×5, 16=2 ⁴ , 17 , 18=2×3 ² , 19 , 20=2 ² ×5, 21=3×7, 22=2×11, 23 , 24=2 ³ ×3, 25=5 ² , 26=2×13, 27=3 ³ , 28=2 ² ×7, 29 , 30=2×3×5, 31 , 32=2 ⁵ , 33=3×11, 34=2×17, 35=5×7, 36=2 ² ×3 ² , 37 , 38=2×19, 39=3×13, 40=2 ³ ×5, 41 , 42=2×3×7, 43 , 44=2 ² ×11, 45=3 ² ×5, 46=2×23, 47 , 48=2 ⁴ ×3, 49=7 ² , 50=2×5 ² , 51=3×17, 52=2 ² ×13, 53 , 54=2×3 ³ , ...

  The factors p _j ^{a _j} are called the prime components of n .

• Goldbach's conjecture : Every even number > 4 is the sum of two odd primes.
• p is prime precisely when p divides ( p -1)!+1
• Mersenne primes and other large primes

• Conway and Guy. The Book of Numbers . chapter 5
• Devlin. All the Math that's Fit to Print . chapters 1, 2, 4, 8, 63, 105, 130.
• Hardy & Wright. Theory of Numbers . section 1.2
• Stepney. Prime Candidates