mersenneforum.org Primes in Decimal Expansions
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 2017-06-13, 18:49 #1 davar55     May 2004 New York City 2×2,099 Posts 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.
 2017-06-15, 14:06 #2 davar55     May 2004 New York City 419810 Posts 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.
 2017-06-26, 14:06 #3 davar55     May 2004 New York City 2×2,099 Posts 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?
 2017-06-27, 06:38 #4 J F     Sep 2013 23×7 Posts '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?
2017-06-27, 12:59   #5
CRGreathouse

Aug 2006

2·2,969 Posts

Quote:
 Originally Posted by J F Dumb question: is there some sort of measure for 'how far away from beeing normal' some number with know properties is?
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.

 2017-06-27, 14:15 #6 davar55     May 2004 New York City 106616 Posts 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.
2017-06-27, 14:42   #7
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

23C616 Posts

Quote:
 Originally Posted by davar55 Perhaps ambiguity in what is expected, perhaps it asks for too much information without providing organization
"That's a bingo! Is that the way you say it?"

 2017-06-27, 15:03 #8 CRGreathouse     Aug 2006 2×2,969 Posts http://mathworld.wolfram.com/ConstantPrimes.html 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.
2017-06-27, 15:43   #9
rogue

"Mark"
Apr 2003
Between here and the

23·751 Posts

Quote:
 Originally Posted by CRGreathouse http://mathworld.wolfram.com/ConstantPrimes.html 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.
Oooh! I see some other potential projects to work on...

I see that MathWorld is not up to date with OEIS for Pi-Primes.

 2017-06-27, 18:52 #10 davar55     May 2004 New York City 419810 Posts Points taken. There may be some worthwhile project within this puzzle, but it remains to be dug out and well defined.
2017-06-27, 20:03   #11
rogue

"Mark"
Apr 2003
Between here and the

177816 Posts

Quote:
 Originally Posted by davar55 Points taken. There may be some worthwhile project within this puzzle, but it remains to be dug out and well defined.
There is no reason that you couldn't create a thread for some of them. You mentioned $sqrt(2)$ and $sqrt(3)$ 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)".

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