Investigating the Diophantine equation [tex]q = \frac{n^2+1}{p}}[/tex] where [tex]{p}[/tex] is a prime number, [tex]n,q[/tex] are integers per definition(adsbygoogle = window.adsbygoogle || []).push({});

The prime numbers can be sorted into two groups

Group 1 has no solution and

Group 2 has the solution [tex] n = \{ a\times p - b{ \ },{ \ } a\times p + b \} {\ \ \ }\forall { \ \ }a>=0[/tex]

The table below list results for the first view primes, there is no particular pattern which divides the primes into either group 1 or 2 nor a pattern for the value [tex]b[/tex] and there seem to be an equal number group1 and group2 primes.

[tex]\begin{array}{cc,c,c}

{No.&Group\ 1&Group\ 2&b\\

1&{}&2&1\\

2&3&{}&{}\\

3&{}&5&2\\

4&7&{}&{}\\

5&11&{}&{}\\

6&{}&13&5\\

7&{}&17&4\\

8&19&{}&{}\\

9&23&{}&{}\\

10&{}&29&12\\

11&31&{}&{}\\

12&{}&37&6\\

13&{}&41&9\\

14&43&{}&{}\\

15&47&{}&{}\\

16&{}&53&23

\end{array}[/tex]

example the for the 10th prime =29 [tex] q= (12^2+1)/29 = 5[/tex]

and 29-12 = 17 [tex] q =(17^2+1)/29 =10[/tex]

and 29+12 = 41 [tex] q =(41^2+1)/29 =58[/tex]

and 2x29-12=46 [tex] q =(46^2+1)/29 =73[/tex]

and 2x29+12=46 [tex] q =(70^2+1)/29 =169[/tex] which is a perfect square.

etc

A further interesting property is that for many (if not all)[tex]p_2[/tex] a prime in Group 2 a infinite number of [tex]a[/tex] exists, such that [tex]\frac{(a\times p_2 \pm b)^2+1}{p_2}}[/tex] is a perfect square. (read [tex]\pm[/tex] as plus or minus b)

47318x29-12=1372210 [tex] q =(1372210^2+1)/29 =64929664969 = 254813^2 [/tex]

My question is - are there other properties that can be attributed to the Group1 or Group2 primes?

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# Prime Numbers in the Diophantine equation q=(n^2+1)/p and p is Prime

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