Prime number 3

 
 

3^4 = 81   has 5 divisors:     1,   3,   9,   27,   81

Sum of divisors:     1 + 3 + 9 + 27 + 81 = 121 = 11^2

 
 

Are there any other primes   p   such that the sum of all positive integer divisors of   p^4   is equal to a square of an integer?

 
 
Solution:

If   p   is a prime, then the sum of all positive integer divisors of   p^4   equals

1 \; + \; p \; + \; p^2 \; + \; p^3 \; + \; p^4

If 1 \; + \; p \; + \; p^2 \; + \; p^3 \; + \; p^4 \; = \; n^2

where   n   is a positive integer.

then we have obviously

(2 \,p^2 + p)^2 \; < \; (2 \,n)^2 \; < \; (2 \,p^2 + p + 2)^2,

and it follows that we must have

(2 \,n)^2 \; = \; (2 \,p^2 + p + 1)^2

Thus,

4 \, n^2 \; = \; 4 \, p^4 \; + \; 4 \, p^3 \; + \; 5 \, p^2 \; + \; 2 \, p \; + \; 1

and since

4 \, n^2 \; = \; 4 \, (p^4 + p^3 + p^2 + 1)

we have

p^2 \; - \; 2 \, p \; - \; 3 \; = \; 0

which implies     p \; | \; 3

hence     p \; = \; 3

If fact, for   p \; = \; 3   we obtain   1 + 3 + 3^2 + 3^3 + 3^4 = 11^2

Thus, there exists only one prime   p,   namely   p \; = \; 3.

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Prime Numbers and tagged , . Bookmark the permalink.

2 Responses to Prime number 3

  1. pipo says:

    Checking all primes up to 100000 and powers 2 up to 9, there is only one other example:
    7^3 = 343
    Sum of its divisors (1+ 7+ 49 + 343) is 400.
    pipo

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