- #1
AntonVrba
- 92
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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
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?
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?