What is the best method for finding integer solutions to a challenging function?

In summary, there is no quick way to find integer values of x that give integer values for y. However, if ##P^2-4R## is a square, Pythagorean triplets can be used as solutions. Otherwise, a brute force approach or more information about the values of P and R is needed.
  • #1
drosser
13
0
Is there a quick way to find integer values of x that give integer values for y?

(x^2-R)/(P-2x)=y

sqrt(R) rounded down<x<P/2

an equivalent equation is

x^2+Px+R=y y= a perfect square

sqrt(x^2+Px+R)= integer

P and R are integer values. They are very large.
P=1.720901664588208977632751606930114527882871349707453690712637328347852193783039275682367157744911327176901e+106

R=1.611966555644167779663503662738501807226651661942209780569274299995114404468640924608971613224013135298666e+105

Maybe a generalized equation or a program?
 
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  • #2
You could search for it in a brute force approach. Otherwise we probably need more information about ##P,R##, e.g. if ##P^2-4R## is a square in which case we have Pythagorean triplets as solutions, or the prime factor decomposition of them.
 

What are integer values of a function?

Integer values of a function refer to the set of whole numbers that can be obtained by inputting different values into the function. These values may include positive and negative integers, as well as zero.

How do you find the integer values of a function?

To find the integer values of a function, you can input a series of values into the function and observe the corresponding output. The resulting set of outputs will be the integer values of the function.

What is the significance of integer values in a function?

Integer values in a function can represent important points on a graph, such as the x-intercepts or turning points. They can also help determine the behavior and shape of the function.

Can a function have an infinite number of integer values?

Yes, a function can have an infinite number of integer values. For example, a linear function will have infinitely many integer values as it continues to extend in both positive and negative directions on the number line.

How do you determine if a function has any integer values at all?

A function will have integer values if there are any input values that produce whole number outputs. To determine this, you can plug in various values into the function and check if the resulting output is a whole number.

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