Estimation

In one sentence

Real numbers form one continuous line, but integers come one step at a time: there is no integer between $3$ and $4$ , and no square between $3^{2}$ and $4^{2}$ .

So for integers, an inequality is sometimes as strong as an equation:

If a quantity is strictly between two consecutive squares, it is not a square.
If you can prove that an unknown is at most some number, try the finitely many cases that are left one by one.

“Estimation” means using inequalities to squash infinitely many possibilities down to a few you can count on your fingers.

Predict first

$n$ is a positive integer. Can $n^{2} + n + 1$ be a perfect square?

Yes, for some small

n

Yes, but only for a very large

n

Choose an answer first; then you can see the result.

Stop 1 · Common methods for Diophantine equations (Vol. 6)

Change one constant and the story changes: this time there are solutions, and they hide where the squeeze fails.

Example: find all positive integer solutions of y² = x² + x + 7

Step 1 / 6

It is close to x²
The right side is a bit bigger than $x^{2}$ . Is it squeezed between $x^{2}$ and the next square, $(x + 1)^{2}$ ? Try $x = 9$ first: $81 < 97 < 100$ , squeezed.

$x^{2} < x^{2} + x + 7 < (x + 1)^{2} ?$
When does the squeeze work?
Every large x is ruled out
6 left: check each one
Finish the check
Conclusion

Stop 2 · Integer solutions of equations (Vol. 2)

When there are no “consecutive squares” to squeeze with, try another idea: put the unknowns in order, and the smallest one cannot be big.

Example: positive integers x ≤ y ≤ z satisfy x + y + z = xyz. Find all solutions

Step 1 / 5

A product grows faster than a sum
When all three numbers are large, $x y z$ is far bigger than $x + y + z$ . So solutions can only involve small numbers. How do you make “small” precise?

$x + y + z = x y z$
Enlarge the sum
Cancel z
Check each case
Conclusion

“Assume $x \leq y \leq z$ ” is not laziness: the equation is symmetric in the three letters, and only after putting them in order is there a “largest” one to bound with.

Stop 3 · The Pythagorean equation (Vol. 6)

Integer right triangles also come “one step at a time”: the hypotenuse $c$ is longer than the leg $b$ , and both are integers, so $c \geq b + 1$ .

Predict first

A right triangle has positive integer sides, and one leg is $9$ . What is the largest possible length of the other leg?

Choose an answer first; then you can see the result.

Example: which integer right triangles have a leg of 9?

Step 1 / 5

Write it as a difference of squares
Let the other leg be $b$ and the hypotenuse $c$ .

$c^{2} - b^{2} (c - b) (c + b) = 81 = 81$
Estimate first: only finitely many
Estimate again: how small is the smaller factor?
Solve
Conclusion

Stop 4 · Quadratic functions and integer roots (Vol. 3)

For a quadratic equation to have integer roots, the discriminant must be a perfect square. The discriminant is a quadratic in the parameter, just right for squeezing between consecutive squares.

Example: k is a positive integer and x² − kx + k = 0 has integer roots. Find k

Step 1 / 6

The discriminant must be a square
The roots are $\frac{k \pm Δ}{2}$ . For them to be integers, $Δ = k^{2} - 4 k$ must be a perfect square.

$Δ = k^{2} - 4 k$
Which square is it closest to?
When does the squeeze work?
Every large k is ruled out
4 left: check each one
Conclusion

All four stops follow the same pattern: first use inequalities to fence off a finite range, then patiently check every case inside it. Neither step can be skipped.

Practice

Exercise 1Basic

The positive integer $n$ satisfies $n^{2} < 200 < (n + 1)^{2}$ . Find $n$ .

Exercise 2Advanced

Find all positive integers $x$ for which $x^{2} + x + 13$ is a perfect square. (Separate them with commas.)

Exercise 3Challenge

Positive integers $x \leq y$ satisfy $x y = x + y + 3$ . Find all possible values of $y$ . (Separate them with commas.)

In one sentence

Stop 1 · Common methods for Diophantine equations (Vol. 6)

Example: find all positive integer solutions of y² = x² + x + 7

Stop 2 · Integer solutions of equations (Vol. 2)

Example: positive integers x ≤ y ≤ z satisfy x + y + z = xyz. Find all solutions

Stop 3 · The Pythagorean equation (Vol. 6)

Example: which integer right triangles have a leg of 9?

Stop 4 · Quadratic functions and integer roots (Vol. 3)

Example: k is a positive integer and x² − kx + k = 0 has integer roots. Find k

Practice

Mistake Museum · Exhibits from this stop

An inequality true only for large numbers, used for all numbers

Bounding from one side only

Stops on this line