Subject: observation about prime numbers (Euclid-style construction)
Davide Rotondo
Subject: A small observation about prime numbers (Euclid-style construction)
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.
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
Key observation
None of the prime numbers used to construct p# divides N.
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.
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.
Conclusion
This implies that the set of prime numbers cannot be finite. Therefore, prime numbers are infinite.
Kind regards,
Davide
Robert Israel
--
You received this message because you are subscribed to the Google Groups "SeqFan" group.
To unsubscribe from this group and stop receiving emails from it, send an email to seqfan+un...@googlegroups.com.
To view this discussion visit https://groups.google.com/d/msgid/seqfan/b98f4e79-a691-48ff-a9bc-23a23e6be5c4n%40googlegroups.com.