The Mystery of Pell and NSW Numbers: Can You Solve It?

In summary, the conversation discusses the problem of finding solutions to the equation 2^n+Q=m^2, where Q=1 and n is odd. The only known solution is 2^3+1=3^2, and the conversation explores the possibility of other solutions and theories related to Pell numbers and NSW numbers. The conversation also presents a proof that (n,p)=(3,3) is the only solution for the equation.
  • #1
T.Rex
62
0
Hi,
I'm looking to solutions of: [tex]2^n+Q=m^2[/tex] , where [tex]Q=1[/tex] .
Obviously, n must be odd.
I already know the trivial solution: [tex]2^3+1=3^2[/tex] and I've started using a naive PARI/gp program for finding (n,m) up to n=59 . No success yet.
Do you know about other solutions or about some theory ?

This is related to Pell numbers [tex](P,Q)=(2,-1)[/tex] and to a series of Prime numbers studied by Newman, Shanks and Williams, called NSW numbers, and generated by: [tex](P,Q)=(6,1)[/tex] .
The idea is to have [tex]D=P^2-4Q=2^n[/tex] and [tex]Q=\pm 1[/tex] .
Since Mersenne numbers are square-free, (2,-1) is the unic solution for Q=-1.
About Q=1, I don't know ...

Tony
 
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  • #2
Well I don't know the solution, but this is how I would look at it. First of all we are looking at odd numbers, although before we do I would like to mention:

[tex]2^0 + 1 = 1^2[/tex]

Right so the from we have is:

[tex]2^{2m + 1} + 1 = 2 \cdot 2^{2m} + 1 = 2 \cdot \left( 2^m \right)^2 + 1 = 2a^2 + 1 = p^2[/tex]

So you are looking for the integer solutions of:

[tex]2 a^2 + 1 = p^2 \quad \text{where:} \quad a = 2^m \quad m, \, p \, \in \mathbb{N}[/tex]

Now, it's not too difficult to deduce from that that p must be odd. Not only that but p - 1 must be twice another square number. Hence:

[tex]p^2 = \left(2q + 1\right)^2 = 4q^2 + 4q + 1[/tex]

So:

[tex]2q^2 + 2q = a^2[/tex]

[tex]q^2 + q = 2^{2m - 1}[/tex]

[tex]q(q + 1) = 2^{2m - 1}[/tex]

Which is a contradiction for q > 1. Now look at q = 1, which is p=3 and your solution. :smile:
 
Last edited:
  • #3
Thanks !

Thanks ! That proves that (n,p)=(3,3) is the unic solution of [tex]2^n+1=p^2[/tex].
Thanks to your proof, I think I have something shorter:
[tex]2^n+1=p^2 \leftrightarrow 2^n=(p-1)(p+1)[/tex] .
In order to have both [tex]p-1=2^\alpha[/tex] and [tex]p+1=2^\beta[/tex], it is clear that there is only 1 solution: [tex]p=3[/tex] .

Thanks !
Tony
 

Related to The Mystery of Pell and NSW Numbers: Can You Solve It?

1. What is the mystery of Pell and NSW numbers?

The mystery of Pell and NSW numbers refers to a mathematical problem involving the Pell equation and numbers from the New South Wales (NSW) lottery. The problem challenges individuals to find a solution to a specific equation using only the numbers drawn in the NSW lottery.

2. What is the Pell equation?

The Pell equation is a type of Diophantine equation, which is an equation with integer solutions. It is written in the form of x2 - Dy2 = 1, where D is a positive integer that is not a perfect square. Solving the Pell equation is a difficult mathematical problem and has been a topic of interest for many mathematicians.

3. How does the NSW lottery numbers relate to the Pell equation?

The NSW lottery numbers are used as the values for D in the Pell equation. This means that the solution to the equation can only use the numbers drawn in the NSW lottery. This adds an extra layer of complexity to the already difficult problem of solving the Pell equation.

4. Has anyone solved the mystery of Pell and NSW numbers?

As of now, the mystery of Pell and NSW numbers remains unsolved. Many mathematicians and individuals have attempted to find a solution, but none have been able to do so. The problem is considered to be a challenging and intriguing mathematical puzzle.

5. Why is the mystery of Pell and NSW numbers important?

The mystery of Pell and NSW numbers is important because it showcases the complexity and beauty of mathematics. It also highlights the importance of problem-solving and critical thinking in the field of science. Additionally, solving the mystery could potentially lead to new discoveries and advancements in mathematics.

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