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# well-ordering principle for natural numbers

Every nonempty set $S$ of natural numbers contains a least element; that is, there is some number $a$ in $S$ such that $a\leq b$ for all $b$ belonging to $S$.

Beware that there is another statement (which is equivalent to the axiom of choice) called the *well-ordering principle*. It asserts that every set can be well-ordered.

Note that the well-ordering principle for natural numbers is equivalent to the principle of mathematical induction (or, the principle of finite induction).

Related:

MaximalityPrinciple, WellOrderedSet, ExistenceAndUniquenessOfTheGcdOfTwoIntegers

Type of Math Object:

Axiom

Major Section:

Reference

## Mathematics Subject Classification

06F25*no label found*65A05

*no label found*11Y70

*no label found*

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## Comments

## general well-ordering principle

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.

## Re: general well-ordering principle

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.

-apk

## Arithmetic Progressions of Prime Numbers

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.

## Re: Arithmetic Progressions of Prime Numbers

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

open.

Kevin

## Re: Arithmetic Progressions of Prime Numbers

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.

## Re: Arithmetic Progressions of Prime Numbers

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