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# prime

An integer $p\in\mathbb{Z}$ is prime if it has exactly two positive divisors. The first few positive prime numbers are $2,3,5,7,11,\dots$.

A prime number is often (but not always) required to be positive.

Prime numbers are very important objects in many fields of mathematics. The notion of prime number has been generalized in a number of ways.

In class field theory, one defines a prime to be a family of equivalent valuations; using this definition, the primes of the rational numbers are given by the positive prime numbers (for a prime $p$, the corresponding valuation is the $p$-adic absolute value) and one extra prime usually denoted $\infty$ (corresponding to the usual absolute value on $\mathbb{Q}$).

In ring theory, one defines the notion of a prime ideal, and also a notion of prime element. However, the notion of prime ideal is more natural in this context than the notion of prime element. For number fields, for example, one has unique factorization of ideals into prime ideals but one does not always have unique factorization of elements into prime elements. The prime ideals of $\mathbb{Z}$ are the ideals generated by the prime numbers, as well as the zero ideal. Note that the ideal generated by $p$ is the same as the ideal generated by $-p$, so we can consider negative primes equivalent to positive primes from this viewpoint as well.

## Mathematics Subject Classification

11A41*no label found*82-00

*no label found*83-00

*no label found*81-00

*no label found*46L05

*no label found*

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## Attached Articles

F\"urstenberg's proof of the infinitude of primes by mathcam

prime numbers, first thousand positive by mps

integer factorization by PrimeFan

proof of infinitude of primes by rspuzio

history of prime numbers by Mathprof

primality by PrimeFan

proof that there are infinitely many primes using the Mersenne primes by PrimeFan

## Comments

## Prime numbers in biology, politics

This morning on "This Week with George Stephanopoulos," George Will pondered how abortion laws would be different if the number of months of gestation of a human baby was a prime number. Sam Donaldson laughed, saying he doesn't know what prime numbers are, to which George Will gave the classic definition "a number divisible only by 1 and itself." (Not that I take George's opinion to be the final say on any topic, political or otherwise).

## Re: Prime numbers in biology, politics

Too bad George was wrong, but that doesn't matter. The cool think is the prime number gestation question. I've heard certain insects like cycadas (spelling?) do so in primes. The idea being that perhaps nature selected for these because parasite that attacked them would emerge on a different cycle and so the optimal survial condition is that the two groups emerge at the same time as few times as possible, so a Chineses remainder theorem problem which of course favors survival of groups with gestation lengths whose GCD is small -- thus primes are best.

However, the flaw with humans is that we don't procreate at exactly the same time of year so prime gestation times wouldn't have a survival angle for us.

## Re: Prime numbers in biology, politics

> I've heard

> certain insects like cycadas (spelling?) do so in primes.

http://en.wikipedia.org/wiki/Cicada#Lifecycle

## Re: Prime numbers in biology, politics

If it was a prime number, we humans would still find a way to make some neat division of the gestation cycle. Let's say 7 months: that would be 28 weeks, which we could then divide into four phases of seven weeks each, or 210 days, which we could divide in three phases of 70 days each (little more than two months). Or say 11 months: that would be about 330 days, which we could divide in three phases of 110 days each (almost four months).

Regardless, there would come a time when the fetus looked so much like a human only the staunchest pro-choicers would still support abortion at that point.

## Re: Prime numbers in biology, politics

> Too bad George was wrong,

I wouldn't be so fast to say George was wrong. Given his profession, the "classic" definition is good enough. Stating the comtemporary definition could have cost him 5 seconds and made the difference between whether or not he got the last word of the roundtable discussion. Though I doubt Sam Donaldson's combover-addled brain could make much sense of either definition.