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well-ordering principle for natural numbers

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06F25 no label found65A05 no label found11Y70 no label found


Maybe we should be specific, and say "well-ordering principle for natural numbers" and "general well-ordering principle", because when you say it, it seems to mean "the naturals can be well-ordered", and when I say it, it means "all sets can be well-ordered." I don't know which is the standard definition.

This once came up in my number theory course; the instructor said "does anyone know what the well ordering principle is?" and I replied with the cop-out "the axiom of choice", and she said "no", and I was baffled. Apparently it was the "the naturals can be well-ordered" version. So yes, someone needs to say something about this.

There exist many examples of (finite) Arithmetic progressions consisting of primes like 5,11,17,23,29 and 61,67,73,79. the longest such example known has 22 primes in arithmetic progression. However, it is not known
if there are arbitrarily long arithmetic progressions of primes.

Markus Frind posted this to the number theory list server back in April:

``Over 10 years ago the only known progression
of 22 primes was found by
Andrew Moran and Paul Pritchard.

AP22 k=0..21
11410337850553 + k*4609098694200 (March 1993)

Today i found a bigger second such instance.

AP22 k=0..21
376859931192959 + k*18549279769020 (April 19th 2003)

The search was conducted over 10 days on a AMD 1800XP
and checked ~20,667,931,547 Potential AP 22's per Second.''

It is known that there are infinitely prime APs of
length 3, but the issue for APs of length 4 remains


Yup Buddy, I knew it. It was proved in '40s (I guess) by S.S.Pillai.
However, we don't know if there are infinitely many arithmetic progressions of three "consecutive" primes. (Like 7,13,19 are primes
in AP but are not consecutive primes as 11 comes between 7 and 13;
and in fact 17 also causesa "problem"). Longest known AP of consecutive primes has length 10.

are these sets of numbers in arithmetic progression also ????????

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