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Is it, or is it not? (Posted on 2007-04-10) Difficulty: 4 of 5
Is (258+1)/5 an integer? A prime?

See The Solution Submitted by Federico Kereki    
Rating: 3.6667 (3 votes)

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Solution Solution To The Problem | Comment 2 of 12 |

2^2(Mod 5) = -1, giving 2^58 (Mod 5) = -1, so that :
(2^58+1)(Mod 5) = 0..(#)

Accordingly, (2^58+ 1)/5 must be an integer.

Now, we observe that:

2^58 + 1

= (2^29 + 1)^2 (2^15)^2

= (2^29 + 2^15 + 1)(2^29 2^15+1)

Now, 2^29 +/- 2^15 +1 > 2^15(2^14+/- 1) + 1 > 2^15> 5
.(##)

Since both the factors of 2^58+1 is greater than 5, it follows from (#) that at least one of the factors in (##) must be evenly divisible by 5. Hence:

(2^58+1)/5= xy, where both x and y must be positive integers, each greater than 1.

Consequently, (2^58 + 1)/5 must be a composite whole number.

Edited on April 11, 2007, 1:07 am
  Posted by K Sengupta on 2007-04-10 13:25:12

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