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, p₁^a₁ ... p_n^a_n, and this decomposition is unique.

  2, 3, 4=2², 5, 6=2×3, 7, 8=2³, 9=3², 10=2×5, 11, 12=2²×3, 13, 14=2×7, 15=3×5, 16=2⁴, 17, 18=2×3², 19, 20=2²×5, 21=3×7, 22=2×11, 23, 24=2³×3, 25=5², 26=2×13, 27=3³, 28=2²×7, 29, 30=2×3×5, 31, 32=2⁵, 33=3×11, 34=2×17, 35=5×7, 36=2²×3², 37, 38=2×19, 39=3×13, 40=2³×5, 41, 42=2×3×7, 43, 44=2²×11, 45=3²×5, 46=2×23, 47, 48=2⁴×3, 49=7², 50=2×5², 51=3×17, 52=2²×13, 53, 54=2×3³, ...

  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