Help!!!

Pls help me solving this problem.

#NumberTheory #Primes #Integers #PrimeNumbers

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`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
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Comments

$N^{2}-1=(N+1)(N-1)$ So no such primes!! (Except when N-1=1)

Similarly for $N^{2}-4=(N+2)(N-2)$ No primes!! (Except when N-2=1, Oh but wait! It implies N=3 and $3^{2}-1=8$ which is not a prime!)

What I think is for all perfect squares a, at most a single prime may exist!

For non perfect squares, I am getting an intuition that infinite primes may exist!

Pranjal Jain - 6 years, 7 months ago

what about ${ n }^{ 2 }-3$

shivamani patil - 6 years, 7 months ago

$4^{2}-3=13$ so it may be a prime number! And I guess maybe ∞ such prime numbers

Pranjal Jain - 6 years, 7 months ago

Well if we talk about your question if we consider $N^{2}$ -2, we will get the primes when N=odd number except 1 and if we consider $N^{2}$ -5 , we will get the primes when N= even number except 2...

In my view this works the best!!!

jaiveer shekhawat - 6 years, 7 months ago

You made a good observation that if $N$ is even, then $N^2 - 2$ is even and hence not a prime if $N > 2$ .

However, this does not imply that if $N$ is odd, then $N^2 - 2$ must be a prime. For example, $11 ^ 2 -2 = 119 = 7 \times 17$ . We can show that if $N \equiv 11 \pmod{14}$ , then $N^2 - 2$ is a multiple of 7 and hence not prime.

Calvin Lin Staff - 6 years, 7 months ago

7*19=133.

shivamani patil - 6 years, 7 months ago

@Shivamani Patil – Yeah! @Calvin Lin Typoed! 119=7×17

Pranjal Jain - 6 years, 7 months ago

@Pranjal Jain – @shivamani patil @Pranjal Jain Thanks! Fixed the typo.

Calvin Lin Staff - 6 years, 7 months ago

well infinitely many prime numbers are of the form 6k±1, it will cover all the primes....

jaiveer shekhawat - 6 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$