[seqfan] Re: A139414
Harry J. Smith
hjsmithh at sbcglobal.net
Thu Jan 29 23:57:14 CET 2009
Neil:
The process of generating primes using polynomials have been in the
mathematics literature since Euler.
Here are some notes from http://math453fall2008.wikidot.com/lecture-2
Generating Primes Through Polynomials
We asked earlier if there was any formula we could use to generate prime
numbers, something which would make discovering new prime numbers as simple
as plug-and-chug.
A long time ago the amazing mathematician Leonard Euler showed that the
polynomial
n^2 + n + 41 (6)
generates distinct prime values for all integral inputs between 0 and 39.
This was an amazing discovery, and even today the record for producing
consecutive primes by a polynomial sits at just 57 (in 2005, the record was
just 43!). So one might ask: is there a polynomial out there which only
takes prime values? Sadly, the answer is know. However..
I think the 4 polynomials in sequence A139414 are unique in how many primes
they generate Sequence A155814 is just a nice companion.
p1(n) =4*x^2 - 146*x + 1373,
p2(n)= 4*x^2 - 144*x + 1459,
p3(n)= 4*x^2 - 142*x + 1301,
p4(n)= 4*x^2 - 140*x + 1877.
I know you cannot please everybody, but I will miss the sequence on
polynomial primes.
Have you seen A090101 Numbers n such that n and the 6 successive integers
yield primes
if substituted for x in polynomial 5x^2+5x+1.
-Harry
> -----Original Message-----
> From: seqfan-bounces at list.seqfan.eu [mailto:seqfan-bounces at list.seqfan.eu]
On
> Behalf Of N. J. A. Sloane
> Sent: Thursday, January 29, 2009 1:31 PM
> To: seqfan at list.seqfan.eu
> Subject: [seqfan] Re: A139414
>
> I agree with those comments! A139414 and its child
> A155814 are too artificial and will be deleted
>
>
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