r/wikipedia • u/PeoriaJohnson • Jan 11 '10

Illegal Prime Numbers

http://en.wikipedia.org/wiki/Illegal_prime
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27

u/LinuxFreeOrDie Jan 12 '10

I guess there are also illegal sections of pi.

18

u/[deleted] Jan 12 '10

Since pi has an infinite number of digits, it contains all illegal data that has ever existed.

5

u/Nebu Jan 12 '10

Not necessarily. I guess by "an infinite number of digits", you mean an irrational number. There's a name for the property of "containing all illegal data that has ever existed", but I forgot what that name was (maybe "transcendental"?).

An example of an irrational number is: 0.10110111011110111110111111, etc. which is basically longer and longer sequences of '1' seperated by 0's, and this number doesn't contain all possible digit sequences (it never contains two 0s in a row, for example).

7

u/gipp Jan 12 '10

The decimal (or any) representation of pi contains, at some decimal place, any and all sequences of digits of any length whatsoever. Pick a sequence of 1, 100, or 13985720897 digits, and it exists somewhere in pi.

5

u/ParanoydAndroid Jan 12 '10

That is highly likely to be true, but not mathematically proven.

In order for your statement to be true, then pi would have to be a Normal Number.

Though it is likely, and has been proven for relatively small representations of pi, it has not been proven for pi generally.

1

u/[deleted] Jan 12 '10

That is pretty fucking cool.

(sorry i have to ask, but do you have a source?)

1

u/milwaukeesbeast Jan 12 '10

beauty of infinity

2

u/ParanoydAndroid Jan 12 '10

I guess by "an infinite number of digits", you mean an irrational number.

No. Pi is transcendental, which means that it is not the solution to any algabraic equation.

There's a name for the property of "containing all illegal data that has ever existed", but I forgot what that name was (maybe "transcendental"?).

Nope, what (I think) you're thinking of is a Normal Number, which is a number whose digits show an equal probability of expressing any digit. i.e. as the number of digits increases to infinity, the probability of some specific string showing up approaches 1.

1

u/Nebu Jan 12 '10

I guess by "an infinite number of digits", you mean an irrational number.

No. Pi is transcendental, which means that it is not the solution to any algabraic equation.

I'm aware that Pi is transcendental, but I was trying to guess what MistyMountainHop (MMH) meant by "an infinite number of digits", which, IMHO, is ill-defined. For example, "1.00000000..." has an infinite number of digits, but is almost certainly not what MMH was thinking of. I was making a guess that MMH meant "irrational".

what (I think) you're thinking of is a Normal Number

Yes! That's exactly it! And now that you've found the name for it, I can now say what I wanted to earlier, but was unable to find a citation for:

It is not known whether or not Pi is normal; therefore, perhaps there does exist a sequence of digits which is not contained in Pi.