Prime Factorization

Every natural number n > 1 has a unique representation as a product of prime numbers, except for the order of the factors. (Fundamental theorem of number theory)

The representation n = p₁^e₁ · p₂^e₂ . . . p_n^e_n  as the product of the prime powers is called the canonical prime factorization.

The program factorizes natural numbers smaller than 10¹⁴  to their prime powers.

Examples:

  99999999999901 = 19001 · 5262880901
  99999999999001 = 107 · 401 · 1327 · 1756309
  99999999990001 = Prime number
 
    3938980639167 = 314 · 77
999330136292431 = 999712 · 99991
    1596644705119 = 909091 · 1756309
	
 100000000000027 = 73² · 271 · 751 · 92203	
 100000000000037 = 1858741 · 53799857	
 100000000000047 = 3 · 7 · 83 · 57372346529	
 100000000000057 = 23 · 4347826086959	
 100000000000067 = Prime number
 100000000000077 = 3 · 17 · 3299 · 594357­
 100000000000087 = 11 · 12647 · 718819411
 100000000000097 = Prime number

                                11 = Prime number
                              101 = Prime number
                            1001 = 7 · 11 · 13
                          10001 = 73 · 137
                        100001 = 11 · 9091
                      1000001 = 101 · 9901
                    10000001 = 11 · 909091
                  100000001 = 17 · 5882353
                1000000001 = 7 · 11 · 13 · 19 · 52579
              10000000001 = 101 · 3541 · 27961
            100000000001 = 112 · 23 · 4093 · 8779
          1000000000001 = 73 · 137 · 99990001
        10000000000001 = 11 · 859 · 1058313049
      100000000000001 = 29 · 101 · 281 · 121499449
    1000000000000001 = 7 · 11 · 13 · 211 · 241 · 2161 · 9091
  10000000000000001 = 353 · 449 · 641 · 1409 · 69857

The first 7 perfect numbers

                                  6 = 2 · 3
                                28 = 22 · 7
                              496 = 24 · 31
                            8128 = 26 · 127
                    33550336 = 212 · 8191
                8589869056 = 216 · 131071
            137438691328 = 218 · 524287                  

Supplement

Instead of a single number, you can also factorize an interval or a sequence of numbers. The input is identical to that of the Sequences and Series  program.

Carl Friedrich Gauss (1777-1855) in the Disquisitiones Arithmeticae (1801):
It is so well known that the problem of distinguishing between the prime numbers and the composite numbers and of breaking them down into their prime factors is one of the most important and useful in all arithmetic, and has occupied the diligence and wisdom of the ancient and modern geometers that there is no need to say much about it.

See also:

Prime Numbers
Wikipedia: Fundamental theorem of arithmetic