Questions about prime numbers?

Question

Why isn't -1 prime?

What is the first prime number formed from the digits of pi. 14, 141, 1415, 14159, ... ?

Is there a formula or easy method to determine how many prime numbers are between a and 2a? a is an integer > 1

Answer 1

1 and -1 are units, and as such can't be prime. Abstract Algebra is the subject that deals with this.
http://mathworld.wolfram.com/AbstractAlgebra.html
http://mathworld.wolfram.com/Unit.htm

14,159 is the first prime in your sequence. (141,592,653,589 is next.)
14 = 2 * 7
141 = 3 * 47
1,415 = 5 * 283

The fastest way is sieving. The fastest methods of sieving are the Sieve of Atkin and the Wheel Sieve.

Paper describing the Wheel Sieve (techncal):
http://citeseer.ist.psu.edu/132206.html

Page describing the Sieve of Atkin (nontechnical):
http://en.wikipedia.org/wiki/Sieve_of_Atkin

Page describing the Sieve of Eratosthenes, a simpler version that isn't as fast as the above:
http://primes.utm.edu/glossary/page.php?sort=SieveOfEratosthenes


By the Prime Number Theorem, you'd expect about x/(ln x) primes between x and 2x for large x. Using bounds due to Pierre Dusart this can be made precise: for x >= 60184, there will be between
2x/(ln(2x)-1)-x/(ln(x)-1.1))
and
2x/(ln(2x)-1.1)-x/(ln(x)-1))
primes between x and 2x. This can probably be improved on without too much work, even though Dusart's bounds are fairly tight taken individually.

Answer 2

If -1 were considered prime, we would no longer have unique factorization of numbers into primes. For example, 5 could be written as both 5 and 5*(-1)*(-1) or even 5*(-1)*(-1)*(-1)*(-1), etc.

It's not fair play to ask three questions at once, but the second question is probably for computer people, and for the third the prime number theorem suggests the number is roughly (a/log(a)), though I'm sure there is not a formula or easy method for an *exact* formula.

Answer 3

* A prime number has to have two natural number (i.e., positive) divisors-- the prime number itself and 1

* 3. .....[14159265357932385...]

*I have no idea. Excluding 2, I think this could give you a pretty good estimate though: ([2a-a]/9) *(2) + (3)

Answer 4

a prime no has only one factor not considering 1,itself.
so 2 has only 2 itself as a factor other than 1.such nos are prime
1 is not a prime no as it is factor of all nos being the multiplicative identity

Answer 5

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Answer 6

No idea... but would love to know the answer myself.