Primes in Decimal Expansions
 

Like in the Primes in Pi thread, a similar sequence can be computed
for primes in other decimal sequences. I suggest it might be interesting
to look for primes in 1/pi, sqrt2, sqrt3, and e, either from the beginning of
the decimal expansions or from within as in prmes in pi.


I want to note that since pi * 1/pi = 1, the semiprimes formed by the
product of a prime in pi and a prime in 1/pi all subsume within the
expansion of 1.

The number of decimal sequences is of course non-denumerably infinite.
Attacking this puzzle includes selecting which sequences to prime-search.

Pi and 1/pi, and integral powers of these are perhaps normal and promising
places to search, possibly due to the relationship of primes and pi.
sqrt2 qnd sqrt3 are the tip of an iceberg containing the integral roots of
all rational numbers.
e and its powers and roots are interesting and "easy" to compute.

Such sequences are easy to define and label.
Perhaps the information about primes in pi and e will help
us determine whether such numbers as e*pi are rational or transcendental.

Are the primes in sqrt2 or sqrt3 comparable in length to those in pi and 1/pi?
All of these are perhaps normal, but does the transcendentalism of pi and 1/pi
affect their "internal primeness" differently from the merely irrational sqrt2 and sqrt3?

'Transcendentalism' shouldn't play a role since that, per se,
says nothing about the distribution of digit values (I think).
For that you have 'Normalism'.

Dumb question: is there some sort of measure for 'how far away
from beeing normal' some number with know properties is?

[QUOTE=J F;462133]Dumb question: is there some sort of measure for 'how far away
from beeing normal' some number with know properties is?[/QUOTE]

If you think that the number is actually normal, but you only know finitely many digits, I'd recommend the chi-squared statistic. If you think it's not normal, then you think there's some k and some k-digit string which appears with a frequency other than 1/b^k; I'd pick the smallest k and one of the strings maximizing the distance from the expected frequency and measure that.

Since no one seems to want to work this problem, there must be some
internal flaw in its presentation. Perhaps ambiguity in what is expected,
perhaps it asks for too much information without providing organization,
perhaps specifying these sequences is too general?

I've seen pi, phi, and e worked on, from the lead digit. Adding a few more
irrational numbers, such as 1 / pi, sqrt2, 1 / sqrt2,, and any other choices,
would extend the implicit data base of primes in such sequences.
Not enough is known about primes within such sequences.

[QUOTE=davar55;462162]Perhaps ambiguity in what is expected,
perhaps it asks for too much information without providing organization
[/QUOTE]
"[I]That's a bingo![/I] Is that the way you say it?"

[url]http://mathworld.wolfram.com/ConstantPrimes.html[/url]
has a nice summary table, you could see if there are differences between the (known) algebraic numbers and the others. It's a little hard to do analysis because not that many terms are known -- it's hard to test big numbers for primality. :smile:

[QUOTE=CRGreathouse;462171][url]http://mathworld.wolfram.com/ConstantPrimes.html[/url]
has a nice summary table, you could see if there are differences between the (known) algebraic numbers and the others. It's a little hard to do analysis because not that many terms are known -- it's hard to test big numbers for primality. :smile:[/QUOTE]

Oooh! I see some other potential projects to work on...

I see that [URL="http://mathworld.wolfram.com/Pi-Prime.html"]MathWorld[/URL] is not up to date with [URL="http://oeis.org/A060421"]OEIS[/URL] for Pi-Primes.

Points taken. There may be some worthwhile project within this puzzle,
but it remains to be dug out and well defined.

[QUOTE=davar55;462193]Points taken. There may be some worthwhile project within this puzzle,
but it remains to be dug out and well defined.[/QUOTE]

There is no reason that you couldn't create a thread for some of them. You mentioned [TEX]sqrt(2)[/TEX] and [TEX]sqrt(3)[/TEX] in your first post. Looking for primes and PRPs in the decimal portion of those number (the string of digits after the decimal point) has also been done for some numbers.

This might be worthy of creating a subforum just for "Primes in (name your decimal)".