A number theory puzzle

**wonderboy1953** · September 15th 2010, 06:10 AM

I give you 7x - 17y = 1. Now give me back how many solution pairs (in natural numbers) there are for the equation? For example (5,2) and (22,9) are solutions (if you can't figure this one out in a week, I'll drop a hint and if you still can't solve it after another week, then I'll solve it for you).

**undefined** · September 15th 2010, 06:19 AM

Originally Posted by wonderboy1953

I give you 7x - 7y = 1. Now give me back how many solution pairs (in natural numbers) there are for the equation? For example (5,2) and (22,9) are solutions (if you can't figure this one out in a week, I'll drop a hint and if you still can't solve it after another week, then I'll solve it for you).

Typo? 7*5 - 7*2 most definitely does not equal 1.

In fact it is easily shown that 7x - 7y = 1 has no integer solutions; the LHS is a multiple of 7 while the RHS is not.

**wonderboy1953** · September 15th 2010, 06:26 AM

Originally Posted by undefined

Typo? 7*5 - 7*2 most definitely does not equal 1.

In fact it is easily shown that 7x - 7y = 1 has no integer solutions; the LHS is a multiple of 7 while the RHS is not.

Correction made.

**undefined** · September 15th 2010, 06:37 AM

Originally Posted by wonderboy1953

I give you 7x - 17y = 1. Now give me back how many solution pairs (in natural numbers) there are for the equation? For example (5,2) and (22,9) are solutions (if you can't figure this one out in a week, I'll drop a hint and if you still can't solve it after another week, then I'll solve it for you).

We have Bezout's identity with a sign change. There are infinite solutions, and a solution exists iff x > 4 and $x\equiv 5\pmod{17}$ .

For $k \ge 0$ we solve for y

7*(17k + 5) - 17y = 1

7*17k + 35 - 17y = 1

17y = 7*17k + 34

y = 7k + 2

So the solution set is given by (x,y) = (5 + 17k, 2 + 7k) where k ranges over the non-negative integers.

**wonderboy1953** · September 15th 2010, 06:56 AM

That's the solution using parametric equations.

**Wilmer** · September 15th 2010, 11:34 AM

Methinks Sir Diophantine just turned over in his grave.

**undefined** · September 15th 2010, 11:38 AM

Originally Posted by Wilmer

Methinks Sir Diophantine just turned over in his grave.

I realize my solving for y could be shortened realizing that

7(x + 17) - 17(y + 7) = 7x - 17y

but what exactly are you referring to?

**wonderboy1953** · September 15th 2010, 12:15 PM

Originally Posted by Wilmer

Methinks Sir Diophantine just turned over in his grave.

Certainly not English.

**Soroban** · September 16th 2010, 06:48 AM

Hello, wonderboy1953!

I have a very elementary solution . . .

$\text{Given: }\;7x - 17y \:=\: 1$

$\text{How many solutions in natural numbers are there for the equation?}$

Answer: there are a brizillian solutions.

Solve for $x\!:\;\;x \:=\:\dfrac{17y+1}{7} \;=\;2y + \dfrac{3y+1}{7}$

Since $\,x$ is a natural number, $(3y+1)$ must be divisible by 7.

We find that: . $y \;=\;2,\,9,\,16,\,\hdots,\,7t-5\;\text{ for }t \in N$

Then: . $x \;=\;2(7t-5) + \dfrac{3(7t-5)+1}{7} \;=\;17t - 12$

Solutions: . $\begin{Bmatrix}x &=& 17t - 12 \\ y &=& 7t-5 \end{Bmatrix}\;\;\text{ for }t \in N$

Of course, this is basicaly undefined's solution.

**Wilmer** · September 16th 2010, 08:46 AM

Originally Posted by undefined

I realize my solving for y could be shortened realizing that
7(x + 17) - 17(y + 7) = 7x - 17y
but what exactly are you referring to?

Nothing in particular UnD; I had just skimmed over:
Diophantine Equations
and saw WB's equation...

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