## PRIME CHAIN OF 3237, 1671 distinct primes

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• This prime chain was not produced with the melded equations method that was used in most of the other prime chain postings on this website, but was rather
Message 1 of 1 , Aug 19 9:07 AM
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This prime chain was not produced with the melded equations method
that was used in most of the other "prime chain" postings on this
website, but was rather found by means of a prime-producing
generator closely akin to Eric Rowland's formula for an infinite
chain: a(1) = 7, n>1, a(n) = a(n-1) + gcd(n,a(n-1) .

It could be argued that neither one is a true prime chain since
there is a vast sea
of 1's in-between each prime, and that mine is not really even
a 'formula' since it
does not achieve an infinite consecutive list, but I see these
objections as
being more semantic than substantial in nature. I also believe that
the entire set of
primes occurring in the sequence represented by x^2 - x + 1 will
eventually appear in the
formula below,(about 1/2 of all primes) but this is yet to be
proved, and it is unclear due
to equipment limitations if the extreme density of primes at the
beginning of the sequence
continues indefinitely or breaks down at some point.

Let f(x) := x^2 - x + 41, x = 1, k = 2, a = 3 .
Repeat indefinitely a two-step process:
x := x + 1,
If GCD( x^2 - x + 41, (x-1)^2 - (x - 1) + 41 - a* k ) >1,
then k := k + 1;

The first 20 values of the sequence that do not equal '1' are:
47, 227, 71, 359, 113, 563, 173, 839, 251,1187,347, 1607,
461,2099,593, 2663, 743,
3299, 911,4007

Many other values for the 'a' variable above, perhaps infinitely
many, (but not all)
may be substituted in the formula and produce long initial prime
chains.
for a = 3 : 3237 consecutive primes,
a = 2 : 736
a = 4 : 817
a = 5 : 858
a = 6 : 161
a = 7 : 1159
a = 8 : 221
a = 9 : 284
a = 10 : 1276 etc.

Many other equations, perhaps infinitely many, (but not all) may be
substituted
for x^2 - x + 1 in the given formula, or a more complicated
development of that formula,
with good results. Example: f(x) := 5*x^2 + 5*x + 1, x = 1,
k = 2, a = 10 : 553 consecutive primes, 276 distinct primes.

Aldrich Stevens
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