- #1
AntonVrba
- 92
- 0
Investigating the Diophantine equation q = \frac{n^2+1}{p}} where {p} is a prime number, n,q are integers per definition
The prime numbers can be sorted into two groups
Group 1 has no solution and
Group 2 has the solution n = \{ a\times p - b{ \ },{ \ } a\times p + b \} {\ \ \ }\forall { \ \ }a>=0
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 b and there seem to be an equal number group1 and group2 primes.
\begin{array}{cc,c,c}<br /> {No.&Group\ 1&Group\ 2&b\\<br /> 1&{}&2&1\\<br /> 2&3&{}&{}\\<br /> 3&{}&5&2\\<br /> 4&7&{}&{}\\<br /> 5&11&{}&{}\\<br /> 6&{}&13&5\\<br /> 7&{}&17&4\\<br /> 8&19&{}&{}\\<br /> 9&23&{}&{}\\<br /> 10&{}&29&12\\<br /> 11&31&{}&{}\\<br /> 12&{}&37&6\\<br /> 13&{}&41&9\\<br /> 14&43&{}&{}\\<br /> 15&47&{}&{}\\<br /> 16&{}&53&23<br /> \end{array}
example the for the 10th prime =29 q= (12^2+1)/29 = 5
and 29-12 = 17 q =(17^2+1)/29 =10
and 29+12 = 41 q =(41^2+1)/29 =58
and 2x29-12=46 q =(46^2+1)/29 =73
and 2x29+12=46 q =(70^2+1)/29 =169 which is a perfect square.
etc
A further interesting property is that for many (if not all)p_2 a prime in Group 2 a infinite number of a exists, such that \frac{(a\times p_2 \pm b)^2+1}{p_2}} is a perfect square. (read \pm as plus or minus b)
47318x29-12=1372210 q =(1372210^2+1)/29 =64929664969 = 254813^2
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 n = \{ a\times p - b{ \ },{ \ } a\times p + b \} {\ \ \ }\forall { \ \ }a>=0
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 b and there seem to be an equal number group1 and group2 primes.
\begin{array}{cc,c,c}<br /> {No.&Group\ 1&Group\ 2&b\\<br /> 1&{}&2&1\\<br /> 2&3&{}&{}\\<br /> 3&{}&5&2\\<br /> 4&7&{}&{}\\<br /> 5&11&{}&{}\\<br /> 6&{}&13&5\\<br /> 7&{}&17&4\\<br /> 8&19&{}&{}\\<br /> 9&23&{}&{}\\<br /> 10&{}&29&12\\<br /> 11&31&{}&{}\\<br /> 12&{}&37&6\\<br /> 13&{}&41&9\\<br /> 14&43&{}&{}\\<br /> 15&47&{}&{}\\<br /> 16&{}&53&23<br /> \end{array}
example the for the 10th prime =29 q= (12^2+1)/29 = 5
and 29-12 = 17 q =(17^2+1)/29 =10
and 29+12 = 41 q =(41^2+1)/29 =58
and 2x29-12=46 q =(46^2+1)/29 =73
and 2x29+12=46 q =(70^2+1)/29 =169 which is a perfect square.
etc
A further interesting property is that for many (if not all)p_2 a prime in Group 2 a infinite number of a exists, such that \frac{(a\times p_2 \pm b)^2+1}{p_2}} is a perfect square. (read \pm as plus or minus b)
47318x29-12=1372210 q =(1372210^2+1)/29 =64929664969 = 254813^2
My question is - are there other properties that can be attributed to the Group1 or Group2 primes?
