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A is ans on solving a we get 1+ n^3 = P^2 so it will be satisfied with n=2 and p=3.. the reason is..p will be alws greater than n..so p will alws be odd no..so its square will be alws odd... so n^3 must be even which is possible with only prime no 2.. so 1+n^3 will bcome odd... so satisfied...

2) on solving eq bcom 8p^2 +1 =m..so I m unable to find any prime no which is 24k+1..where k is any + integer.. so not satisfied

(1) \sqrt[3]{1-p^2}=-n, where n is a prime number. (2) 8p^2+1=m, where m is a prime number.

(1) \sqrt[3]{1-p^2}=-n OR p^2=1+n^3=(1+n)(n^2-n+1) If n=2, p=3 If n>2, then n is odd, (n+1) is even, hence p^2 is even, but this is only possible if p=2 and if p=2, n is cube_root(-3) which is not an integer So only possibility is p=3 Sufficient

(2) 8p^2+1=m p is an integer and a prime so it can either be of the form 3k+1 or 3k-1 or be 3 (all other numbers of form 3k are composite) 8*(3k+1)^2+1=8*(9k^2+6k+1)+1=3*(24k^2+16k+3) .. which cannot be a prime since k>=1 8*(3k-1)^2+1=8*(9k^2-6k+1)+1=3*(24k^2-16k+3) .. which cannot be a prime since k>=1 8*3^2+1=73 is a prime So only possible value of p is 3 Sufficient

(1) \sqrt[3]{1-p^2}=-n, where n is a prime number. (2) 8p^2+1=m, where m is a prime number.

I wrote this question, so below is my solution:

(1) \sqrt[3]{1-p^2}=-n, where n is a prime number.

It's clear that p\neq{2} as in this case \sqrt[3]{1-p^2}\neq{integer}, so p is a prime more than 2, so odd. Then 1-p^2=odd-odd=even --> cube root from even is either an even integer or not an integer at all, we are told that \sqrt[3]{1-p^2}=-n=integer, so -n=even=-prime --> n=2 (the only even prime) --> \sqrt[3]{1-p^2}=-2 --> p=3. Sufficient.

(2) 8p^2+1=m, where m is a prime number.

p can not be any prime but 3 for 8p^2+1 to be a prime number: because if p is not 3, then it's some other prime not divisible by 3, but in this case p^2 (square of a not multiple of 3) will yield the remainder of 1 when divided by 3 (not a multiple of 3 when squared yields remainder of 1 when divided by 3), so p^2 can be expressed as 3k+1 --> 8p^2+1=8(3k+1)+1=24k+9=3(8k+3) --> so 8p^2+1 becomes multiple of 3, so it can not be a prime. So p=3 and in this case 8p^2+1=73=m=prime. Sufficient.

I am really confused. Can someone please explain me,

Statement 2:

I understand that if we work out the 8P^2 + 1 : where P = 2 we get m = 33, which is NOT a prime.

Great! I am on-board till here.

Is the strategy to try every prime number from 2, onwards??

How is the value of P chosen??

You state P cannot be any prime but 3 for 8P^2 + 1 to be a prime number. How can you be so sure? I do not have an exhaustive list of all prime number, but this claim is little far fetched.

I am really confused. Can someone please explain me,

Statement 2:

I understand that if we work out the 8P^2 + 1 : where P = 2 we get m = 33, which is NOT a prime.

Great! I am on-board till here.

Is the strategy to try every prime number from 2, onwards??

How is the value of P chosen??

You state P cannot be any prime but 3 for 8P^2 + 1 to be a prime number. How can you be so sure? I do not have an exhaustive list of all prime number, but this claim is little far fetched.

Please clear my confusion.

There are infinitely many prime numbers so there is no "list of all primes".

Next, p can not be any prime but 3 for 8p^2+1 to be a prime number: because if p is not 3, then it's some other prime not divisible by 3, but in this case p^2 (square of a not multiple of 3) will yield the remainder of 1 when divided by 3 (not a multiple of 3 when squared yields remainder of 1 when divided by 3), so p^2 can be expressed as 3k+1 --> 8p^2+1=8(3k+1)+1=24k+9=3(8k+3) --> so 8p^2+1 becomes multiple of 3, so it can not be a prime. So p=3 and in this case 8p^2+1=73=m=prime.

OK. But I am still not clear about why is it important to be divisible by 3.

"because if p is not 3, then it's some other prime not divisible by 3, but in this case p^2 (square of a not multiple of 3) will yield the remainder of 1 when divided by 3 (not a multiple of 3 when squared yields remainder of 1 when divided by 3)"

Bunuel wrote:

There are infinitely many prime numbers so there is no "list of all primes".

Next, p can not be any prime but 3 for 8p^2+1 to be a prime number: because if p is not 3, then it's some other prime not divisible by 3, but in this case p^2 (square of a not multiple of 3) will yield the remainder of 1 when divided by 3 (not a multiple of 3 when squared yields remainder of 1 when divided by 3), so p^2 can be expressed as 3k+1 --> 8p^2+1=8(3k+1)+1=24k+9=3(8k+3) --> so 8p^2+1 becomes multiple of 3, so it can not be a prime. So p=3 and in this case 8p^2+1=73=m=prime.

I guess, you will have to explain this a little more.

OK. But I am still not clear about why is it important to be divisible by 3.

"because if p is not 3, then it's some other prime not divisible by 3, but in this case p^2 (square of a not multiple of 3) will yield the remainder of 1 when divided by 3 (not a multiple of 3 when squared yields remainder of 1 when divided by 3)"

I guess, you will have to explain this a little more.

Because if p IS NOT divisible by 3 then 8p^2+1 IS divisible by 3 and thus can not be a prime, so p must be divisible by 3, only prime divisible by 3 is 3.

Tough one, I could solve the question by plugging in the numbers but didn't think of trying (especially for 2nd statement) the above concepts, despite knowing for a number to be prime it should be either 6k+1 or 6k-1. Similarly, for statemet1 despite knowing that p can not be 2 and it would be more than didn't think that it must be odd and apply Odd & Even concept here.

Thank you! for reminding me to use concepts rather than just guessing and then getting doubtful solutions (and doing silly mistakes).

_________________

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