 ## Proof of the Beal Conjecture through the Fundamental Theorem of Arithmetic

### Submission history

Sign In. Forums Contests Search Help. Like this:. Any integer greater than 1 is either a prime number , or can be written as a unique product of prime numbers ignoring the order.

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A Prime Number is a number that cannot be exactly divided by any other number except 1 or itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, Yes, 2 , 3 and 7 are prime numbers, and when multiplied together they make Any of the numbers 2, 3, 4, 5, 6, Suppose , and suppose.

I must show.

## Proof of the Beal Conjecture through the Fundamental Theorem of Arithmetic

If , then , which contradicts. By the preceding theorem,. This establishes the result for. Suppose that.

### Explanations (2)

Grouping the terms, I have. By the case , either or.

If , I'm done. Otherwise, if , then p divides one of , , In either case, I've shown that p divides one of the 's, which completes the induction step and the proof.

## Fundamental theorem of arithmetic - Wikipedia

Fundamental Theorem of Arithmetic First, I'll use induction to show that every integer greater than 1 can be expressed as a product of primes. If n is prime, I'm done.

Otherwise, n is composite, so I can factor n as , where. By induction, a and b can be factored into primes. Then shows that n can, too. Here the p's are distinct primes, the q's are distinct primes, and all the exponents are greater than or equal to 1.

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