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 September 24th, 2007, 05:31 PM #1 Senior Member   Joined: Dec 2006 Posts: 1,111 Thanks: 0 An unusual prime number question Suppose that we took all the prime numbers (in whatever base we were working with) and wrote them all down in a row, with a decimal point at the beginning. In other words: 0.23571113171923... Would the resulting number be irrational (and thus non-constructable)? Would it also be transcendental?
 September 24th, 2007, 07:36 PM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms This is the Copeland–Erdős constant. It is base-10 normal, and as such irrational. I don't think its transcendence has been proved, but surely no one doubts it -- think about the consequences of showing that it's algebraic!
September 25th, 2007, 05:28 AM   #3
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 think about the consequences of showing that it's algebraic!
Wow, I didn't realize that somebody has already tried that! (Then again, people have tried just about everything relating to prime numbers.) If it were shown to be an algebraic constant, would that mean that the prime numbers could be generated somehow using an algebraic formula? If it were proven to be transcendental, would that mean that the prime number sequence was non-algebraic, and thus that there is no formula for prime numbers?

September 25th, 2007, 05:46 AM   #4
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 Originally Posted by Infinity Wow, I didn't realize that somebody has already tried that! (Then again, people have tried just about everything relating to prime numbers.) If it were shown to be an algebraic constant, would that mean that the prime numbers could be generated somehow using an algebraic formula? If it were proven to be transcendental, would that mean that the prime number sequence was non-algebraic, and thus that there is no formula for prime numbers?
Yes and no. Clearly if it was algebraic one could reverse-engineer it to get the primes. But there are many, many formulas for the prime numbers, and if this isn't one there are still plenty of others. How else could I list 2, 3, 5, 7, 11, ...? There are even a number of closed form formulas for the primes, using just sigma notation and basic functions.

September 25th, 2007, 07:51 AM   #5
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 How else could I list 2, 3, 5, 7, 11, ...?
Oh, I see. So you need the primes to get the constant, so there's no point in reverse engineering the constant to get the primes. When people talk about the magic holy grail formula for prime numbers, are they typically referring to a formula whose output could predict the next n prime numbers, in order, and also whose output consisted of nothing but prime numbers?

September 25th, 2007, 08:02 AM   #6
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 Originally Posted by Infinity Oh, I see. So you need the primes to get the constant, so there's no point in reverse engineering the constant to get the primes.
That's not what I said, but it's true enough.

Quote:
 Originally Posted by Infinity When people talk about the magic holy grail formula for prime numbers, are they typically referring to a formula whose output could predict the next n prime numbers, in order, and also whose output consisted of nothing but prime numbers?
There are functions of that sort already. Most of them are too slow to be useful.

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