The Prime Debate: Is 1 prime?

[Urban Ranger]

Originally posted by geeslaka

Originally posted by Urban Ranger
Of course it is, because 1^Infinity evaluates to 1, nothing more.

Factorisation is the process of finding the complete set of numbers that when multiplied together, will evaluate to the number being factored.
Thus, evaluating part of the factorisation defeats the purpose.

Factorisation continues until all non-trivial terms are discovered. Since 1 is by definition trivial, I cannot see where your argument comes from.

[geeslaka]

Factorisation continues until all non-trivial terms are discovered. Since 1 is by definition trivial, I cannot see where your argument comes from.

The non-trivial part is there purely for convenience.

[connorkimbro]

the monkey says, 1 is prime! and you know the monkey must be smart, because he's wearing glasses.

[geeslaka]

Are you sure that's a monkey? I've seen old guys that look like that. Specifically my grand father.

[Jaguar]

[Urban Ranger]

Originally posted by geeslaka

Factorisation continues until all non-trivial terms are discovered. Since 1 is by definition trivial, I cannot see where your argument comes from.

The non-trivial part is there purely for convenience.

You disregard the non-trivial part because adding it in makes absolutely no difference to the end result.

By your logic, any additions and subtractions cannot be considered complete since you get to play with an infinite number of 0's.

[geeslaka]

You disregard the non-trivial part because adding it in makes absolutely no difference to the end result.

True in some cases, but it does make a difference when determining uniqueness of prime factorisations.

By your logic, any additions and subtractions cannot be considered complete since you get to play with an infinite number of 0's.

True. What is wrong with this ability?

[Kuciwalker]

It is irrelevant whether or not one is "prime". The useful definition of the word prime is the one that does not include the number one, because there are an extraordinary number of properties of prime numbers that fail if you include the number one.

[geeslaka]

It is irrelevant whether or not one is "prime". The useful definition of the word prime is the one that does not include the number one, because there are an extraordinary number of properties of prime numbers that fail if you include the number one.

Name a few of those properties. And this definition is still a matter of convenience.

[Boris Godunov]

Originally posted by geeslaka
Found a statement of the FTA in Enigma_Nova's link.

The
FDA states that every positive integer greater than one can be
expressed uniquely as a product of primes, apart from the
rearrangement of terms.

I think it's great the Food and Drug Administration is finally starting to get a handle on these mathematical issues.

[geeslaka]

I simply ignored that typo as not worthy of notice.

[SnowFire]

Wrong wrong wrong, folks. 1^infinity might be 1. It might not be. It's indeterminate, much like 0/0. The 0/0 case is obvious: 6/2 is cleary 3 because you need to add 3 2's together to get 6. How many 0's do you have to add together to get 0? Well, if you don't add any (0), that works. But if you add 5 0's, you get 0 too. In fact, if you add negative 10 billion 0's, you also get 0. This is why when you get 0/0 as the limit of some function, you have no clue what the answer is and need to use more advanced techniques. Outside of a limit, 0/0 is simply indeterminate.

It may be harder to understand why 1^infinity is the same. But basically, when you multiply a number infinite times, things get wonky. Don't forget that a number infinitely close to 1 is still "1" (just as .9999... = 1), but if you do infinite multiplications on it, you might be able to "undo" that closeness and bring out some larger number.

Here's a webpage I found with some more examples:

http://www.sosmath.com/calculus/indforms/intro/intro.html

So 1^infinity is not a good option for prime factorizations, since that "factorization" might not multiply to the number we think it does.

[Kuciwalker]

Originally posted by geeslaka

It is irrelevant whether or not one is "prime". The useful definition of the word prime is the one that does not include the number one, because there are an extraordinary number of properties of prime numbers that fail if you include the number one.

Name a few of those properties. And this definition is still a matter of convenience.

Why is the number one not a prime?

The number one is far more special than a prime! It is the unit (the building block) of the positive integers, hence the only integer which merits its own existence axiom in Peano's axioms. It is the only multiplicative identity (1.a = a.1 = a for all numbers a). It is the only perfect nth power for all positive integers n. It is the only positive integer with exactly one positive divisor. But it is not a prime. So why not? Below are four answers, each more technical than its precursor.

[b]Answer One: By definition of prime[bb]

The definition is as follows:

[i]An integer greater than one is called a prime number if its only positive divisors (factors) are one and itself.[i]

Clearly one is left out, but this does not really address the question "why?"

Answer Two: Because of the purpose of primes.

The formal notion of primes was introduced by Euclid in his study of perfect numbers (in his "geometry" classic The Elements). Euclid needed to know when an integer n factored into a product of smaller integers (a nontrivially factorization), hence he was interested in those numbers which did not factor. Using the definition above he proved:

The Fundamental Theorem of Arithmetic

Every positive integer greater than one can be written uniquely as a product of primes, with the prime factors in the product written in order of nondecreasing size.

Here we find the most important use of primes: they are the unique building blocks of the multiplicative group of integers. In discussion of warfare you often hear the phrase "divide and conquer." The same principle holds in mathematics. Many of the properties of an integer can be traced back to the properties of its prime divisors, allowing us to divide the problem (literally) into smaller problems. The number one is useless in this regard because a = 1.a = 1.1.a = ... That is, divisibility by one fails to provide us any information about a

Answer Three: Because one is a unit.

Don't go feeling sorry for one, it is part of an important class of numbers call the units (or divisors of unity). These are the elements (numbers) which have a multiplicative inverse. For example, in the usual integers there are two units {1, -1}. If we expand our purview to include the Gaussian integers {a+bi | a, b are integers}, then we have four units {1, -1, i, -i}. In some number systems there are infinitely many units.
So indeed there was a time that many folks defined one to be a prime, but it is the importance of units in modern mathematics that causes us to be much more careful with with the number one (and with primes).

Answer Four: By the Generalized Definition of Prime.

There was a time that many folks defined one to be a prime, but it is the importance of units and primes in modern mathematics that causes us to be much more careful with the number one (and with primes). When we only consider the positive integers, the role of one as a unit is blurred with its role as an identity; however, as we look at other number rings (a technical term for systems in which we can add, subtract and multiply), we see that the class of units is of fundamental importance and they mustbe found before we can even define the notion of a prime. For example, here is how Borevich and Shafarevich define prime number in their classic text "Number Theory:"

An element p of the ring D, nonzero and not a unit, is called prime if it can not be decomposed into factors p=ab, neither of which is a unit in D.

Sometimes numbers with this property are called irreducible and then the name prime is reserved for those numbers which when they divide a product ab, must divide a or b (these classes are the same for the ordinary integers--but not always in more general systems). Nevertheless, the units are a necessary precursors to the primes, and one falls in the class of units, not primes.

[geeslaka]

I cede to the more advanced mathematical knowledge.
SnowFire is correct because infinity does not have to be a whole number and I was not considering that case.

Thank you skywalker.

[Kuciwalker]

You're welcome. Google is useful, n'est-ce pas?