Shobhit said:If $p$ is an odd prime expressible as
$$p=x^2+5y^2$$
where $x,y$ are integers, then prove that $p \equiv 1 \text{ or }9\text{ (mod 20)}$.
mathbalarka said:How does (2) follows from (1)?
mathbalarka said:Ah, I see, I have interpreted the question wrong. I interpreted is as : p is an odd prime expressible as x^2 + 5y^2 if and only if p is 1 or 9 modulo 20.
But perhaps OP must have intended exactly this? Can Shobhit clarify whether he did a typo or not?
Shobhit said:Balarka, the question is correct. You had interpreted it correctly.
"Primes of the form x² + 5y²" are prime numbers that can be written as the sum of a perfect square and five times another perfect square. In other words, they can be expressed in the form x² + 5y², where both x and y are integers.
"Primes of the form x² + 5y²" are a subset of all prime numbers and have a specific mathematical form. Regular prime numbers can be expressed in various forms and do not necessarily follow a specific pattern.
The study of "Primes of the form x² + 5y²" is important in number theory and has connections to other areas of mathematics, such as algebraic geometry and elliptic curves. It also has practical applications in cryptography and coding theory.
The Fibonacci sequence is closely related to "Primes of the form x² + 5y²". In fact, every prime number of the form x² + 5y² is either a member of the Fibonacci sequence or can be expressed as a sum or difference of two consecutive Fibonacci numbers.
Yes, there are several unsolved problems related to "Primes of the form x² + 5y²". One notable example is the Bunyakovsky conjecture, which states that the polynomial x² + 1 always produces infinitely many primes when substituted with integer values. This conjecture has been proven for certain values of y, but remains open for the case of y = 5.