MHB Are Both Solutions to the Diophantine Equation $x^4 - 2y^2 = 1$ Valid?

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The discussion centers on the Diophantine equation $x^4 - 2y^2 = 1$, with participants exploring its solutions. The primary argument is that the equation can be factored, leading to the conclusion that for integer solutions, $x$ must be odd, specifically of the form $x = 2k + 1$. The analysis reveals that both $k$ and $k + 1$ must be perfect squares, ultimately identifying the solutions as $(1, 0)$ and $(-1, 0)$. Participants also discuss the implications of negative integers and the uniqueness of prime factorization in the context of the equation. The conversation highlights the necessity of considering both positive and negative solutions to fully resolve the problem.
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Hey! (Smirk)

I am looking at the following exercise:

Prove that the diophantine equation $ x^4-2y^2=1 $ has only two solutions.

It is: $$(x−1)(x+1)(x^2+1)=2y^2 $$

If $x=2k \Rightarrow 2y^2=(2k-1)(2k+1)(4k^2+1), \text{ that is not possible, because } 2 \nmid (2k-1)(2k+1)(4k^2+1)$

So,it is $x=2k+1$.

Replacing this,we get:

$$4k(k+1)(2k^2+2k+1)=y^2$$

So, $k,k+1,2k^2+2k+1$ must all be a square.
$$ k=a^2 $$
$$k+1=a^2+1=b^2 $$
$$b^2−a^2=1⇒a=0,b= \pm 1 \Rightarrow k=0 $$
Therefore, $x=1$.I found only the solution: $(1,0)$ .. So,have I done something wrong? (Thinking) (Thinking)
 
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Shouldn't (-1,0) be a solution as well?
 
Deveno said:
Shouldn't (-1,0) be a solution as well?

Yes,but I haven't found this solution with the way that I did it. (Doh)
So,do I have to solve the exercise in an other way? (Thinking)
 
"So, k,k+1,2k^2+2k+1 must all be a square."

I don't see an apparent reason for that.
Like Deveno said, (-1,0) is another solution, then with x=2k+1 follows k=-1,
and -1 is not a square...
 
evinda said:
Hey! (Smirk)

I am looking at the following exercise:

Prove that the diophantine equation $ x^4-2y^2=1 $ has only two solutions.

It is: $$(x−1)(x+1)(x^2+1)=2y^2 $$

If $x=2k \Rightarrow 2y^2=(2k-1)(2k+1)(4k^2+1), \text{ that is not possible, because } 2 \nmid (2k-1)(2k+1)(4k^2+1)$

So,it is $x=2k+1$.

Replacing this,we get:

$$4k(k+1)(2k^2+2k+1)=y^2$$

So, $k,k+1,2k^2+2k+1$ must all be a square.

I don't see how you draw this conclusion. Yes, they are all co-prime, but -1 divides any integer, and is not prime. What happens if $k = -1$?
 
Deveno said:
I don't see how you draw this conclusion. Yes, they are all co-prime, but -1 divides any integer, and is not prime. What happens if $k = -1$?

Taschee said:
"So, k,k+1,2k^2+2k+1 must all be a square."

I don't see an apparent reason for that.
Like Deveno said, (-1,0) is another solution, then with x=2k+1 follows k=-1,
and -1 is not a square...

Sice, $4k(k+1)(2k^2+2k+1)=y^2$

doesn't it mean that $\exists a,b,c \text{ such that } k=a^2, k+1=b^2 \text{ and } 2k^2+2k+1=c^2$ ? Or am I wrong? (Sweating)
 
evinda said:
Hey! (Smirk)

I am looking at the following exercise:

Prove that the diophantine equation $ x^4-2y^2=1 $ has only two solutions.

It is: $$(x−1)(x+1)(x^2+1)=2y^2 $$

If $x=2k \Rightarrow 2y^2=(2k-1)(2k+1)(4k^2+1), \text{ that is not possible, because } 2 \nmid (2k-1)(2k+1)(4k^2+1)$

So,it is $x=2k+1$.

Replacing this,we get:

$$4k(k+1)(2k^2+2k+1)=y^2$$

So, $k,k+1,2k^2+2k+1$ must all be a square.
$$ k=a^2 $$
$$k+1=a^2+1=b^2 $$
$$b^2−a^2=1⇒a=0,b= \pm 1 \Rightarrow k=0 $$
Therefore, $x=1$.I found only the solution: $(1,0)$ .. So,have I done something wrong? (Thinking) (Thinking)
you have found

$k , k+1, 2k^2 + 2k +1$ must all be square

you also can have
( alternatively)
$- k , -(k+1), 2k^2 + 2k +1$ must all be square which gives the second solution
 
Put another way, we can factor $c^2 = a^2b^2$ two ways:

$c = a^2b^2 = (-a^2)(-b^2)$.

Prime factorization is only unique "up to units" and the integers have TWO units: 1 and -1. People often overlook the second unit.
 
evinda said:
Sice, $4k(k+1)(2k^2+2k+1)=y^2$

doesn't it mean that $\exists a,b,c \text{ such that } k=a^2, k+1=b^2 \text{ and } 2k^2+2k+1=c^2$ ? Or am I wrong? (Sweating)
Not necessarily.
You need to prove, that k, k+1 and 2k^2+2k+1 are coprime:
k and k+1 are obviously coprime, if p|k, then p|2k^2 and p|2k,
so p \nmid 2k^2+2k+1;
If p|k+1, then p|2(k+1)^2 = (2k^2+2k+1) + (k+1) + k
Suppose p|2k^2+2k+1 , then you have p|k in contradiction to p|k+1

Then you still have the possibility that two of those terms are negativ squares,
i.e. k=-a^2, k+1=-b^2;
 

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