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# Infinite Primes

Pro Problems > Math > Logic > Proofs > Indirect Proofs

## Infinite Primes

Use an indirect proof to show that there are infinitely many prime numbers.

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Problem by Mr. Twitchell

## Solution

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## Similar Problems

### Sum of Trig Functions

Show that if θ is between 0º and 90º inclusive, then sin θ + cos θ ≥ 1. Use an indirect proof.

### Harmonic Proof

Create an indirect proof for the following statement: The harmonic mean of any two positive numbers lies inclusively between the two numbers.

### X, Y, and Z proof

Prove by contradiction: for any integers x, y, and z with z > 1, x2 - yz2 ≠ z.

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### Irrational Fraction

Let a and b be positive integers, with b not a perfect square. Show that the following expression is irrational, using an indirect proof.

a +
b
a -
b

Hint: Remember that an irrational number is a number which cannot be written as the quotient of two integers.

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