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Not a Prime (Posted on 2012-12-20) Difficulty: 3 of 5
Prove that 5100+575+550+525+1 is not a prime number.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 3.0000 (2 votes)

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Some Thoughts answer | Comment 2 of 6 |
Without factoring I do not know of many primality tests.
In addition to the test presented by Charlie in the former post, one test is to test the digital root. If the digital root of the number is 3 (except for the prime number 3 itself), 6, or 9, then the number is not prime.

The digital roots of 5^n (beginning with n=1) is the period 6 sequence: {5,7,8,4,2,1}.
100 modulo 6 is 4, digital root for such is thus 4;
 75 modulo 6 is 3, digital root for such is thus 8;
 50 modulo 6 is 2, digital root for such is thus 7;
 25 modulo 6 is 1, digital root for such is thus 5;
The digital root of the given expression, then, is [4 + 8 + 7 + 5 (+ 1) = 25 -> 2 + 5] = 7. Thus it does not fail this primality test, and may at this point be yet prime.

5^100 = 7888609052210118054117285652827862296732064351090230047702789306640625;
5^75 = 26469779601696885595885078146238811314105987548828125;
5^50 = 88817841970012523233890533447265625;
5^25 = 298023223876953125;

5^100 + 5^75 + 5^50 + 5^25 + 1 =
7888609052210118080587065254524747981434984467341564893722534179687501

The number has the prime factors: 3597751; 28707251; 4032808198751; 767186663625251; and 24687045214139234043375683501. Thus it itself is not prime.

The value of the expression above was calculated using a big number calculator and the factors were found using WIMS' Factoris factoring utility.

Edited on December 21, 2012, 9:26 am
  Posted by Dej Mar on 2012-12-20 22:20:50

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