- #1
robert Ihnot
- 1,059
- 1
Let x, y, n all represent positive integers in x^4+nY^4. It seem there is a lot of primes in this set. In fact, even allowing x=1, n=1, we look at 1+Y^4, we see pairs, y=1, f(y)=2, (2,17), (4,257), (6,1297), (16,65537), (20,160001) Possibly an infinite set?
Take the case of x=1, n=2, giving (1,3) now we have a problem since the form 1+2Y^4, will be divisible by 3 unless we take y as a multiple of 3 giving primes: (3,163), (6, 2593), (18,209953). In the case of x=1, n=3, 1^4+3*4^4 =769, (6, 3889),(8,12289)
We can continue with this, increasing n, but in the case of x^4+4y^4 there is only one prime solution, x=1, y=1, F(x,y) = 5.
QUESTION: Is there an n such that X^4+nY^4 NEVER gives a prime number?
Take the case of x=1, n=2, giving (1,3) now we have a problem since the form 1+2Y^4, will be divisible by 3 unless we take y as a multiple of 3 giving primes: (3,163), (6, 2593), (18,209953). In the case of x=1, n=3, 1^4+3*4^4 =769, (6, 3889),(8,12289)
We can continue with this, increasing n, but in the case of x^4+4y^4 there is only one prime solution, x=1, y=1, F(x,y) = 5.
QUESTION: Is there an n such that X^4+nY^4 NEVER gives a prime number?
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