@Shivamani Patil
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\(4^{2}-3=13\) so it may be a prime number! And I guess maybe ∞ such prime numbers
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Pranjal Jain
·
2 years, 2 months ago

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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!!!
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Jaiveer Shekhawat
·
2 years, 3 months ago

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@Jaiveer Shekhawat
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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.
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Calvin Lin
Staff
·
2 years, 2 months ago

## Comments

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TopNewest\(N^{2}-1=(N+1)(N-1)\)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

mayexist!For non perfect squares, I am getting an intuition that infinite primes may exist! – Pranjal Jain · 2 years, 2 months ago

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– Shivamani Patil · 2 years, 2 months ago

what about \({ n }^{ 2 }-3\)Log in to reply

– Pranjal Jain · 2 years, 2 months ago

\(4^{2}-3=13\) so it may be a prime number! And I guess maybe ∞ such prime numbersLog in to reply

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 · 2 years, 3 months ago

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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 · 2 years, 2 months ago

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– Shivamani Patil · 2 years, 2 months ago

7*19=133.Log in to reply

@Calvin Lin Typoed! 119=7×17 – Pranjal Jain · 2 years, 2 months ago

Yeah!Log in to reply

@shivamani patil @Pranjal Jain Thanks! Fixed the typo. – Calvin Lin Staff · 2 years, 2 months ago

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well infinitely many prime numbers are of the form 6k±1, it will cover all the primes.... – Jaiveer Shekhawat · 2 years, 3 months ago

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