Question 1

why is 1 not a composite or prime number

Accepted Answer

Mathematicians now consider 1 to not be prime because it is inconvenient if it is prime. Believe it or not, back in the 19th century some mathematicians did consider 1 to be prime.

If 1 is prime it breaks the *Fundamental Theorem of Arithmetic* which says that all  numbers >1 have a unique prime factorization. (The prime factorizations would no longer be unique if 1 was prime)
e.g. 
If 1 is not prime the prime factorization of 6 is 6=2*3, and it is unique.
However, if 1 is prime then we could also have the following prime factorizations of 6: 6=1 * 2 * 3, 6=1 * 1 * 2 * 3, 6=1 * 1 * 1 * 2 * 3, etc. The prime factorization of 6 would no longer be unique.

The *Fundamental Theorem of Arithmetic* is a useful tool in mathematics, so mathematicians decided to exclude 1 from being prime. The biggest benefit to this, is that 1 doesn't have to be considered as a special case every time we prove something (This saves a lot of paper and even more headaches).

Hope this makes sense

Question 2

A "prime/no prime" button _would_ be pretty nice to have, now that I think about it...the calculator is capable of computing some pretty advanced calculations, why can't it tell if a number is prime? Even if it takes a while, it would be useful to know...

Accepted Answer

While a prime/not prime button would be cool, there are some logistical difficulties with putting it on a calculator. Most primality testing algorithms in use today are based on probabilistic principles. For example, the Miller-Rabin Primality test can definitively tell you that a number is not prime, but can only tell you with very high probability if a number is actually prime. The reason is that the algorithm (which is an amazingly elegant piece of mathematics) tests a given input for "strong pseudoprimality" i.e., it checks whether your input satisfies a series of conditions when paired with a randomly chosen number, like 2 or 3. If you run the algorithm 50 times with 50 random numbers, then the probability that your number (of less than 200 digits) is prime is greater than 99.99%. So you might ask: is there a completely deterministic test for primality? That was discovered recently by two undergraduates (and their advisor) in 2000. It answered a fundamental question, but the algorithm is way too slow to be of any practical use. So coming back to your calculator, they usually feature deterministic algorithms, and primality testing, at least at a basic level, does not conform to that.

Question 3

ok i really dont understand prime numbers cant any one help me?

Accepted Answer

A prime number is a number that can only be divided by only the number one and itself.
Such as: 7 is a prime number because it can only be divided by 1 and 7.

Question 4

But my calculator _does_ have an equivalent function for primality tests. It is actually factorization. I used to use CASIO fx-350ES PLUS, and when I click SHIFT-degree (º ' "), which on the above calculator is actually "←," I get the factorization of the number currently in the answer section. If the number is prime, the calculator displays the number in parentheses. (By the way, for 1337 I get 7×191; and when I factor 1 or a non-positive integer or fraction, I get a Math ERROR.)

On a TI-Nspire CX, I press menu-2(Number)-3(Factor), or I type in "factor(" and I get the factoring function. It can also "factor" negative numbers and fractions, and when it can't factor a number (1, 0, decimals, irrationals, imaginary numbers) it displays the original number (simplified).

In fact, the calculator in the picture is the first scientific or graphing calculator I've seen that definitely doesn't have a factoring function. Of course, It's probably because I never look closely at other people's calculators.

Accepted Answer

Good point! Many modern scientific calculators have a "factor" function, and usually have since the early 90s. In fact, many of them will actually test to see if a number is prime before factoring it. For example, my trusty HP 50g will mark the number 99,194,853,094,755,497 as prime immediately, but if I try to factor 99,194,856,159,718,057 (the nearest semiprime), it runs for about a minute and a half before giving up.

The main point that the lesson is this, though: it's a lot easier to multiply primes to get a number, than it is to factor a number into its primes! I hope the next two lessons make that clear.

The main point that the lesson is this, though: it's a lot easier to multiply primes to get a number, than it is to factor a number into its primes!  I hope the next two lessons make that clear.

Question 5

Are negative numbers prime or composite? Are their structures similar to positive numbers?

Accepted Answer

That's a good question!  Negative numbers are neither prime nor composite.  (The same applies to 0.)

The question of "is it prime or composite" only applies to numbers greater than 1.  It doesn't apply to 0 or 1, or negative numbers, or fractions, or ducks, or airplanes.

Negative numbers do have a similar structure to positive numbers; for example, a negative number (less than -1) can always be factored into -1 times some primes.

At this level, mathematians start dividing numbers into several different kinds.  They talk about the kinds of numbers that can count how many things there are (which are called cardinal numbers; there's 0, 1, 2, or 3 apples), or positions in a list of things (called ordinal numbers; this apple is the first, second, third, etc); cardinals and ordinals are the same kinds of numbers until you deal with infinities.  They start talking about kinds of numbers that can be fractions of integers (rationals), or kinds of numbers that can talk about the differences of two other numbers (integers).  That last kind of numbers is where negative numbers come in.  Talking about negative numbers doesn't make sense when you're talking about cardinals or ordinals; you can't have negative three apples!

When mathematicians started talking about these different kinds of numbers, they also realized that some definitions only apply to certain kinds of numbers.  So talking about a prime number meant you were only talking about positive (or "natural") numbers; it didn't make any sense to talk about negative primes, or fractional primes, or anything like that.

So that's a great question: negative numbers aren't prime, and aren't composite; they're a different type of number than those definitions apply to.  And their structure is similar to, but different than, positive numbers.

Question 6

*What is the biggest prime number?*

Accepted Answer

Say we have found the biggest prime number... let us call it P.
Then let us take P! and subtract one. Since P! is divisible by all numbers up to P, P! - 1 cannot be divisible by any of the numbers up to P excluding 1. So we have either found a prime number bigger than P or there is another prime number largest than P that divides into P! - 1. Either way, our statement that P is the largest prime number is contradictory. And therefore we have proved by contradiction that the "biggest prime number" does not exist. Primes continue on forever.

Question 7

I am making a program that you input a number and it tells you whether it is prime or composite. Can you find some big primes for me to test with?

Accepted Answer

Try these: http://www.bigprimes.net/archive/prime/14000000/

Question 8

Why is 2 the only even prime number?

Accepted Answer

By definition, any even number larger than 2 will be divisible by 2. Thus it cannot be prime.

Course: Computer science theory > Unit 2

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