![]() ![]() The conjecture that every even number (greater than or equal to 6) can be written as the sum of two odd prime numbers. Our assumption that Pn is the biggest prime has led us to a contradiction, so this assumption must be false, so there is no biggest prime. ![]() This means that either Q must be prime or Q must be divisible by primes that are larger than Pn. This is because the rest of the numbers ending with 5 are divisible by 5 itself. No prime number greater than 5 ends in a 5. 1 is the only even prime number from 1 to 100 Prime numbers are infinite. Now we can see that if we divide Q by any of our n primes there is always a remainder of 1, so Q is not divisible by any of the primes.īut we know that all positive integers are either primes or can be decomposed into a product of primes. Here are twin prime numbers from 1 to 100: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73). Now we can form the number Q by multiplying together all these primes and adding 1, so If we assume that there are just n primes, then the biggest prime will be labelled Pn. We can number all the primes in ascending order, so that P1 = 2, P2 = 3, P3 = 5 and so on. The proof works by showing that if we assume that there is a biggest prime number, then there is a contradiction. Euclid Euclid's Proof that the Set of Prime Numbers is Endless ![]()
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