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Algebraic factorizations

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For positive integer
n > 1
, we have
anbn  =  (ab)
n
i   = 1
  
an  − i bi  − 1
 =  (ab) (an  − 1 + an  − 2 b + an  − 3 b 2 + + a bn  − 2 + bn  − 1).
For odd positive integer
2 n + 1, n > 1,
we have
a 2 n +1 + b 2 n +1  =  (a + b)
2 n +1
i   = 1
  
a 2 n +1 − i (−b)i  − 1
 =  (a + b) (a 2 na 2 n  − 1 b + a 2 n  − 2 b 2 + a b 2 n  − 1 + b 2 n ).
For even positive integer
2 n
, we have
a 2 n + b 2 n  =  (an
2  2 anbn
+ bn ) (an +
2  2 anbn
+ bn ).

Examples

Table of algebraic factorizations
n
an  −  bn,

an + bn.
2

3

4

5

6

7

8

9

10

11

12

Algebraic factorization of integers

629 = 625 + 4 = 25 2 + 2 2 = (25  − 
2  2  ×  25  ×  2
+ 2) (25 +
2  2  ×  25  ×  2
+ 2) = 17  ×  37.
117 = 125  −  8 = 5 3  −  2 3 = (5  −  2) (5 2  +  5  ×  2 + 2 2 ) = 3  ×  39.
133 = 125  +  8 = 5 3  +  2 3 = (5  +  2) (5 2  −  5  ×  2 + 2 2 ) = 7  ×  19.
629 = 625 + 4 = 5 4 + 2 2 = 5 4 + (
2  2
 ) 4 = (5 2  − 
2  2
 ×  5  × 
2  2
+ (
2  2
 ) 2 ) (5 2 +
2  2
 ×  5  × 
2  2
+ (
2  2
 ) 2 ) = (25  −  10 + 2) (25 + 10 + 2) = 17  ×  37.
390629 = 390625 + 4 = 5 8 + 2 2 = 5 8 + ( 
4  2
 ) 8 = (5 4  − 
2  2
 ×  5 2  ×  ( 
4  2
 ) 2 + ( 
4  2
 ) 4 ) (5 4 +
2  2
 ×  5 2  ×  ( 
4  2
 ) 2 + ( 
4  2
 ) 4 ) = (625  −  50 + 2) (625 + 50 + 2) = 577  ×  677.

See also

  • Repunits (base
    b
    ) and repdigits (base
    b
    ) (when the number of digits is composite, we have algebraic factorizations)