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Sieve of Eratosthenes: primes up to 50

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Primes up to 50

The first 15 primes up to 47. There are 15 primes below 50.

Primzahl	#	Primzahl	#	Primzahl	#
2	1	19	8	37	12
3	2	23	9	41	13
5	3	29	10	43	14
7	4	31	11	47	15
11	5	37	12	53	16
13	6	41	13	59	17
17	7	43	14	61	18

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Euclid's proof: infinitely many primes

Assume finitely many primes: p₁, p₂, …, pₙ

N = p₁·p₂·…·pₙ + 1 → N is divisible by none of p₁…pₙ

So N is prime or has a prime factor not in the list — contradiction. ∴ infinitely many primes. QED (Euclid, ~300 BC)

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Primzahl	#	Primzahl	#	Primzahl	#
2	1	19	8	37	12
3	2	23	9	41	13
5	3	29	10	43	14
7	4	31	11	47	15
11	5	37	12	53	16
13	6	41	13	59	17
17	7	43	14	61	18

Primzahl	#	Primzahl	#	Primzahl	#
2	1	19	8	37	12
3	2	23	9	41	13
5	3	29	10	43	14
7	4	31	11	47	15
11	5	37	12	53	16
13	6	41	13	59	17
17	7	43	14	61	18

Primzahl	#	Primzahl	#	Primzahl	#
2	1	19	8	37	12
3	2	23	9	41	13
5	3	29	10	43	14
7	4	31	11	47	15
11	5	37	12	53	16
13	6	41	13	59	17
17	7	43	14	61	18