Exploring the Difference of Two Squares: A Scientific Analysis

[um0123]

Homework Statement

1) Find all pairs of natural numbers whose squares differ by 75.

2) Find all pairs of natural numbers whose squares differ by 79.

3) Prove that there can only be 1 pair of numbers with a prime number difference

Homework Equations

none

The Attempt at a Solution

from common sense i can conclude that the answer to question 1 is: 5 and 10. But i need an equation of some sort to get question 2.

and I am not exactly sure what question 3 is asking.

 

[PhaseShifter]

hint for part 3:
$$n=x^{2}-y^{2}=(x+y)(x-y)$$
where n is some prime number.

Does that make sense?

Then try part 2. Isn't 79 a prime number?

 

[um0123]

mhm, i had already figured that out. but i solved for x and then substituted it into the equation, to get the number and i got 79 = 79, or 75 = 75, depending on the problem i was working out.

 

[PhaseShifter]

What is the definition of a prime number? Look at the part on the right.

 

[um0123]

$$79 = (x+y)(x-y)$$

but how do i solve for one variable so i can substitute. when i do, it cancels each other out.

EDIT: would it be a trick question? since primes can only be multiplied by 1 and their self. whereas 75 is not prime (3*25).

 

[HallsofIvy]

You know that $x^2- y^2= (x- y)(x+ y)= 79$. Further you know that x and y are positive integers so x- y and x+ y are also whole numbers. How many pairs of positive integers are there that multiply to give 79? Set one of such a pair equal to x- y, the other equal to x+ y and solve the two equations.

How many pairs of positive integers are there that multiply to give 75? No, it's not a "trick" question.

 

[um0123]

theres only 1 pair that can make 79 since its prime. 1 and 79.

if you set 1 and 79 as y and x you get

$$79 = (79+1)(79-1)$$

$$79 = (80)(78)$$

which isn't correct.

I know there is something that you are telling me that i just don't understand.

 

[njama]

$$x^2-y^2=79$$

$$y=\sqrt{x^2-79}$$

What is the range of the function y?

 

[um0123]

$$x^2-y^2=79$$

$$y=\sqrt{x^2-79}$$

What is the range of the function y?

i tried that, when i substituted it for y i got:

$$79 = x^2 -(\sqrt{x^2 - 79})^2$$

$$79 = x^2 - x^2 -79$$

$$79=-79$$

 

[Matthollyw00d]

theres only 1 pair that can make 79 since its prime. 1 and 79.

if you set 1 and 79 as y and x you get

$$79 = (79+1)(79-1)$$

$$79 = (80)(78)$$

which isn't correct.

I know there is something that you are telling me that i just don't understand.

You're not quite understand what was said. Don't set 79=x and y=1. Set (x-y)=79 and (x+y)=1, then the reverse, (ie. (x-y)=1 and (x+y)=79).

 

[um0123]

You're not quite understand what was said. Don't set 79=x and y=1. Set (x-y)=79 and (x+y)=1, then the reverse, (ie. (x-y)=1 and (x+y)=79).

OH, because we are finding the squares we need to find the what numbers equal the factors, i get it.

$$x-y=1$$
$$y = -x - 1$$

$$x+y = 79$$
$$y = -x + 79$$

$$x+y = 1$$
$$y=-x + 1$$

$$x-y=79$$
$$y=x-79$$

but what do i do with these values?

 

[Matthollyw00d]

OH, because we are finding the squares we need to find the what numbers equal the factors, i get it.
Case 1:
$$x-y=1$$
$$y = -x - 1$$

$$x+y = 79$$
$$y = -x + 79$$

Case 2:
$$x+y = 1$$
$$y=-x + 1$$

$$x-y=79$$
$$y=x-79$$

but what do i do with these values?

You know have 2 sets of systems of equations each solvable.

 

[um0123]

and when you solve it you get 40 and 39.

and $$40^2 - 39^2 = 79$$

AWESOME!

and also

$$79 = (40-39)(40+39)$$

PERFECT. Thanks so much!