Subject: observation about prime numbers (Euclid-style construction)

[Davide Rotondo]

Subject: A small observation about prime numbers (Euclid-style construction)

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Dear all,

I would like to share a simple observation about prime numbers that I found interesting. It is a variant in the spirit of Euclid’s classical argument for the infinitude of primes.

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Construction

Let

p# = p1 · p2 · ... · pn

be the product of the first n prime numbers (the so-called primorial).

Now define:

N = 10 * p# + 1

For example:

• if p# = 30, then N = 301

• if p# = 210, then N = 2101

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Key observation

None of the prime numbers used to construct p# divides N.

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Proof

Let q be any prime factor of p#.

Then:

• q divides p#

• therefore q also divides 10 * p#

Now assume, for contradiction, that q also divides N = 10 * p# + 1.

Then q would divide the difference:

(10 * p# + 1) − (10 * p#) = 1

Hence q divides 1, which is impossible for any prime number.

Therefore, no prime factor of p# can divide N.

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Consequence

Every prime factor of N must be a prime number not contained in the original list of primes used to construct p#.

Thus, starting from any finite list of prime numbers, we can construct a number that produces at least one new prime not in the list.

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Conclusion

This implies that the set of prime numbers cannot be finite. Therefore, prime numbers are infinite.

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Kind regards,
Davide

[Robert Israel]

Why bother with the 10?

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