Persplexing prodigius Primes

Number Theory Level 3

How many values of " p p " are their such that ( p ) (p) , ( p + 2 ) (p+2) & ( p + 4 ) (p+4) are all primes?

If you think that there are infinite values,enter answer as 20121 20121 .


The answer is 1.

Harsh Shrivastava by Harsh Shrivastava , 20, India

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2 solutions

Vijay Simha
Nov 1, 2016

Unless 3 divides p (in which case p = 3, p + 2 = 5, p + 4 = 7 are all prime), We have p = 3k + 1 or p = 3k + 2. In the first case p + 2 = 3k + 3 is divisible by 3 and cannot be prime since k > 0, in the second case p + 4 = 3k + 6 > 3 is divisible by 3 and cannot be prime.

So there is only one set of primes of the form p, p+2, p + 4.

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Ch Nikhil
Nov 28, 2014

There is only one solution- 3 since a prime can only be of the form (3k+1) , therefore p+2 will always be divisible by 3 .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

there are 2 values of P i.e- 1,3 2 solutions

Suvojit Das - 6 years, 6 months ago

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No. 1 1 is not a prime number.

Satvik Golechha - 6 years, 6 months ago

1 is not a prime. And if p = 2, 4 and 6 are both composite numbers.

Devin Ky - 5 years, 11 months ago

11 is equal to 3k +2 for k= 3.This implies that your solution is wrong. Actually, a right solution is 3 satisfy the result. If p is another prime p = 3k+1 or p =3k+2 for an integer k, If p=3k+1 then p+2 is a multiple of 3(p+2=3(k+1)) and if p=3k+2 then p+4=3(k+2) and it is also multiple of 3

Guillermo Templado - 5 years, 9 months ago

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