## A prime rich sequence???

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• 3 = 2 + 1 5 = 2^2 + 1 7 = 2^2 + 2 + 1 11 = 2^3 + 2 + 1 13 = 2^3 + 2^2 + 1 17 = 2^4 + 1 3 + 1 = 2 * 2 3^2 + 1 = 2 * 5 3^2 + 3 + 1 = 13 = prime 3^3 + 3 + 1 = 31
Message 1 of 2 , Nov 29, 2006
3 = 2 + 1
5 = 2^2 + 1
7 = 2^2 + 2 + 1
11 = 2^3 + 2 + 1
13 = 2^3 + 2^2 + 1
17 = 2^4 + 1

3 + 1 = 2 * 2
3^2 + 1 = 2 * 5
3^2 + 3 + 1 = 13 = prime
3^3 + 3 + 1 = 31 = prime
3^3 + 3^2 + 1 = 37 = prime
3^4 + 1 = 2 * 41

4 + 1 = 5 = prime
4^2 + 1 = 17 = prime
4^2 + 4 + 1 = 3 * 7 ; 2^2 + 2 + 1 = 7
4^3 + 4 + 1 = 3 * 23 ; 2^3 + 2 + 1 is neither 23 nor 3.
4^3 + 4^2 + 1 = 3^4
4^4 + 1 = 257 = prime

5+1 = 2 * 3
5^2 + 1 = 2 * 13
5^2 + 5 + 1 = 31 = prime
5^3 + 5 + 1 = 131 = prime
5^3 + 5^2 + 1 = 151 = prime

6 + 1 = 7 = prime
6^2 + 1 = 37 = prime
6^2 + 6 + 1 = 43 = prime
6^3 + 6 + 1 = 223 = prime
6^3 + 6^2 + 1 = 11 * 23 ; 2^3 + 2^2 + 1 = 11

Does the prime richness extend to larger numbers, or is this an illusion
of small primes?

Kermit < kermit@... >
• I don t think these are particularly good prime-generating polynomials you ve found. Perhaps they are quite good given the smallness of the coefficients,
Message 2 of 2 , Nov 30, 2006
I don't think these are particularly good prime-generating polynomials
you've found. Perhaps they are quite good given the smallness of the
coefficients, though! Most of the examples below have much larger
coefficients than just 0 and 1.

For some really excellent ones, take a look at
http://mathworld.wolfram.com/Prime-GeneratingPolynomial.html
http://www.maa.org/editorial/mathgames/mathgames_07_17_06.html
or toward the bottom of
http://euler.free.fr/contest/PGPReport.htm
where they give polynomials both with integer and rational coefficients,
and which give lots of consecutive primes,
or which give a relatively high proportion of primes in some range
like x = 0 to 1000 ...

--Joshua Zucker
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