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April 11th, 2017, 02:50 AM   #1
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Pythagorean prime in every prime twin

Hi,
I have a thread to understand, but I have some problem with it.

I must prove that in every prime twin there is one and only one Pythagorean prime.

I found something about it on 5 page of:

www.fq.math.ca/Scanned/24-2/sternheimer.pdf

Can anyone explain it a bit easier?

Thank you a lot.

Last edited by skipjack; April 12th, 2017 at 04:44 AM.
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April 12th, 2017, 02:21 AM   #2
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A Pythagorean Prime is of the form $4n + 1$ so the best you can do is to have $2$ consecutive Pythagorean Primes differ by $4$ , like $13$ and $17$

Twin primes differ by $2$ exactly so it is impossible to have $2$ Pythagorean Primes in any set of twin primes $(p_1 , p_2)$


Last edited by agentredlum; April 12th, 2017 at 02:39 AM. Reason: fixed error
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April 12th, 2017, 02:35 AM   #3
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Any prime greater than $2$ must be of the form $4n + 1$ or $4n- 1$

Consider the arbitrary twin prime ordered pair $(p_1, p_2)$ with $p_1 < p_2$ and $2$ cases

Case 1

If $p_1$ is of the form $4n + 1$ then $p_2$ must be of the form $4n + 3$ so only $1$ Pythagorean Prime in this case

Case 2

If $p_1$ is of the form $4n- 1$ then $p_2$ is of the form $4n + 1$ so only $1$ Pythagorean Prime in this case

This exhausts all possible cases for twin primes given your conditions

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