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

• AntonVrba
In summary, Group 2 prime numbers have a solution where n=\{a\times p - b\}. 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.
AntonVrba
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.
$$\begin{array}{cc,c,c} {No.&Group\ 1&Group\ 2&b\\ 1&{}&2&1\\ 2&3&{}&{}\\ 3&{}&5&2\\ 4&7&{}&{}\\ 5&11&{}&{}\\ 6&{}&13&5\\ 7&{}&17&4\\ 8&19&{}&{}\\ 9&23&{}&{}\\ 10&{}&29&12\\ 11&31&{}&{}\\ 12&{}&37&6\\ 13&{}&41&9\\ 14&43&{}&{}\\ 15&47&{}&{}\\ 16&{}&53&23 \end{array}$$

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?

Have you tried looking at your answers modulo some number? mod 2, mod 3, mod 4, or mod 8 often tell you something interesting about integer equations involving squares.

Hurkyl said:
Have you tried looking at your answers modulo some number? mod 2, mod 3, mod 4, or mod 8 often tell you something interesting about integer equations involving squares.
Interesting - all Group2 primes have remainder 1 when divided by 4

AntonVrba said:
Interesting - all Group2 primes have remainder 1 when divided by 4

Except 2.

You can say more and assert that every prime congruent to 1 mod 4 is in your group 2. You're just asking for what primes p does the equation $$n^2\equiv -1\ \mod\ p$$ have a solution n, or when is -1 a square mod p. Look up the Legendre symbol, quadratic residues,Euler's criteria etc.

unique identifiers for Group 2 primes

Let a(n) = n^2 +1. Let p, q be primes from group 2 and P, Q be the unique numbers less than p/2 or q/2, respectively, such that a(P) equals 0 mod p and a(Q) equals 0 mod q.
A. If a(n) equals 0 mod p then n equals either +/- P mod p.
B. If a(P) is composite, i.e. = p*d*q (q is prime, d >/= 1) then all other prime factors of a(P) correspond to still smaller numbers Q such that a(Q) equals 0 mod q. An example is a(12) equals 0 mod 29. Since 12 < 29/2, then 12 is the lowest positive number n such that a(n) = 0 mod 29. The other prime factor of a(12) is 5 which corresponds to q = 5 where Q=2 and 12= 2 mod 5.
C. The P, Q numbers etc and the corrresponding primes (->1 means that all prime factors were previously listed) for n < 16 are
1->2
2->5
3->1
4->17
5->13
6->37
7->1
8->1
9->41
10->101
11->61
12->29
13->1
14->197
15->113

Last edited:
Ramsey - 100% correct - this helps me further

## 1. What is a Diophantine equation?

A Diophantine equation is a type of polynomial equation in which the unknowns are required to be integers. It is named after the ancient Greek mathematician Diophantus.

## 2. What is the significance of prime numbers in the Diophantine equation q=(n^2+1)/p?

In this specific Diophantine equation, the prime number p is used as a divisor to find the value of q, which must also be a prime number. This equation has been studied for centuries and has connections to other areas of mathematics, such as number theory and algebraic geometry.

## 3. What is the importance of prime numbers in mathematics?

Prime numbers play a crucial role in number theory and cryptography. They have unique properties and are the building blocks for all other numbers. They are also used in various real-world applications, such as computer algorithms and data encryption.

## 4. Are there any known solutions for the Diophantine equation q=(n^2+1)/p?

Yes, there are known solutions for this equation, such as (n=1, p=2) and (n=2, p=5). However, it is still an open problem in mathematics to find a general solution for all values of n and p.

## 5. How do prime numbers relate to the Goldbach conjecture?

The Goldbach conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers. This conjecture is closely related to the Diophantine equation q=(n^2+1)/p, as it involves finding solutions for prime numbers in a specific equation. However, the Goldbach conjecture is still unproven and remains one of the most famous unsolved problems in mathematics.

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