Solve Diophantine Equation: $m^2 - n^2 = 105$

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  • Thread starter Jkawa
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In summary, the solutions to the equation $m^2 - n^2 = 105$, where $m$ and $n$ are integers, can be found by identifying the factors of 105 and using them to solve for $m$ and $n$ using the equation $(m-n)(m+n) = 105$. This involves finding the mean of each pair of factors and subtracting the smaller factor from the mean, then plugging these values into the equation to find the solutions. It is important to consider both positive and negative pairs of factors when finding solutions.
  • #1
Jkawa
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Find all solutions to $m^2 - n^2 = 105$, for which both $m$ and $n$ are integers.
 
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  • #2
Re: Help needed for problem

Jkawa said:
Find all solutions to $m^2 - n^2 = 105$, for which both $m$ and $n$ are integers.

Notice $\displaystyle \begin{align*} m^2 - n^2 = \left( m - n \right) \left( m + n \right) \end{align*}$. So what are the factors of 105 that could be possible candidates for m-n and m+n?
 
  • #3
Re: Help needed for problem

Prove It said:
Notice $\displaystyle \begin{align*} m^2 - n^2 = \left( m - n \right) \left( m + n \right) \end{align*}$. So what are the factors of 105 that could be possible candidates for m-n and m+n?

Factors of 105 are 1, 3, 5, 7, 15, 21, 35, 105. I'm sort of confused on the use of these though. I don't see a combination that could satisfy the equation.
 
  • #4
Let's start with the pair $1\cdot105$.

First, compute the mean:

\(\displaystyle \frac{1+105}{2}=53\)

Now, find the difference between the mean and the smaller factor:

\(\displaystyle 53-1=52\)

Thus:

\(\displaystyle (53+52)(53-52)=105\)

Proceed in like manner for the remaining pairs. :)
 
  • #5
Upon inspection, I only see that the only solution to this is $m = 53$ and $n = 52$ because the rest of the factors do not equal 105. Am I correct?
 
  • #6
Jkawa said:
Upon inspection, I only see that the only solution to this is $m = 53$ and $n = 52$ because the rest of the factors do not equal 105. Am I correct?

No, you can do the same with $3\cdot35$.

\(\displaystyle \frac{3+35}{2}=19\)

\(\displaystyle 19-3=16\)

\(\displaystyle (19+16)(19-16)=105\)

Try the other pairs. :D
 
  • #7
MarkFL said:
No, you can do the same with $3\cdot35$.

\(\displaystyle \frac{3+35}{2}=19\)

\(\displaystyle 19-3=16\)

\(\displaystyle (19+16)(19-16)=105\)

Try the other pairs. :D

Oh my calculation was completely off, I was following the exact same algorithm as you posted lol. Thanks!
 
  • #8
Don't neglect the negative pairs of integer factors, for example:

\(\displaystyle (-1)(-105)\)

\(\displaystyle \frac{(-1)+(-105)}{2}=-53\)

\(\displaystyle -53-(-1)=-52\)

\(\displaystyle (-53+(-52))(-53-(-52))=105\)
 

FAQ: Solve Diophantine Equation: $m^2 - n^2 = 105$

What is a Diophantine equation?

A Diophantine equation is a polynomial equation in two or more unknowns where the solutions are restricted to be integers. In other words, the solutions to a Diophantine equation must be whole numbers.

How do you solve a Diophantine equation?

There is no general method for solving all Diophantine equations. However, there are specific techniques that can be used depending on the form of the equation. Some common techniques include factoring, substitution, and using modular arithmetic.

What is the specific Diophantine equation m^2 - n^2 = 105?

The equation m^2 - n^2 = 105 is an example of a difference of squares Diophantine equation. It can be rewritten as (m + n)(m - n) = 105. This means that the solutions must be two integers whose product is 105.

What are the solutions to m^2 - n^2 = 105?

There are multiple solutions to this equation. Some examples include m = 15 and n = 12, m = 55 and n = 50, and m = 106 and n = 105. There are an infinite number of solutions, as long as m and n are integers and their product is 105.

Can a computer be used to solve Diophantine equations?

Yes, computers can be used to solve Diophantine equations. There are algorithms and computer programs specifically designed to solve these types of equations. However, the complexity of the equation and the size of the unknowns can greatly affect the time it takes for a computer to find a solution.

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