## Nice Prime!

June 21st, 2012 | Categories: general math, mathematica | Tags:

In a recent tweet, Cliff Pickover told the world that
7272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727­27272727 is prime. That’s a nice looking prime and it took my laptop 1/100th of a second to confirm using Mathematica 8.

PrimeQ[727272727272727272727272727272727272727272727272727272727272727\
272727272727272727272727272727272727] // AbsoluteTiming

Out[1]= {0.0100000, True}

Can anyone else suggest some nice looking primes?

1. Unfortunately, Mathematica isn’t able to tell me what value of x for Prime[x] will yield that number. (And why aren’t Prime and PrimePi parallelizable, anyway??) It would be awesome if that value was also a neat looking prime.

2. Is 135791113151719212325272931333537394143454749515355575961636567697173757779818385878991939597 nice enough? The interesting thing is that four nontrivial prefixes of it are also primes. (I used isprime(parse(cat(“”, seq(1 .. 97, 2)))) to verify its primality in Maple by the way.)

3. Also, 1003005007009011013015017019021023025027029031033035037039041043045047049051053055057059061063065067069071073075077079081083085087089091093095097099101103105107109111113115117119121123125127129131133135137139141143145147149151153155157159161163165167169171, obtainable as parse(cat(seq(sprintf(“%03d”, i), i=1..171, 2))).

4. There appears to be a serious base 10 bias in this blog post and comments. Does it mean anything for a prime to be “nice looking” outside of a particular base?

5. Yep, they are definitely nice enough! pretty amazing that we can check primality of such large numbers so quickly.

Thanks!

6. @simon..it’s a good point, we are all being very baseist

7. Are these verifications absolute or probabilistic?

8. All Primes… :)

7

727

72727

727272727

72727272727272727

72727272727272727272727272727272727272727272727272727272727272727272727

727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727

727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727

9. 1 and 7 are cool too.

1717171717171717171717171717171

1717171717171717171717171717171717171

10. Why not use only 1 and 7?

1717171717171717171717171717171

1717171717171717171717171717171717171

11. 3 and 7;

373

373737373737373737373

373737373737373737373737373

373737373737373737373737373737373737373737373737373737373737373737373737373737373

…..

and with 1893 numbers

373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373737373

12. 9 and 7

997

99797979797979797979797979797979797979797979797979797979797

997979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797

99797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797

I’m a math prime spammer? Sorry :( I’m ashamed…

13. This is even more improbable! 7 and 8! 4 times better than 7 and 2!

78787

787878787878787878787

787878787878787878787878787

78787878787878787878787878787878787878787878787878787878787878787878787878787878787878787878787

14. @Thales I think we are going to have to come up with a compact notation. On my browser, these numbers scroll right outside the boundaries of the box

15. ‘PrimeQ first tests for divisibility using small primes, then uses the Miller-Rabin strong pseudoprime test base 2 and base 3, and then uses a Lucas test.’

It’s a probabilistic procedure. More details at http://mathworld.wolfram.com/Rabin-MillerStrongPseudoprimeTest.html

16. Poor MATLAB, cannot handle such huge number :(… though I don’t understand why.

—–
Core i5, RAM 4GB, MATLAB R2012a 64bit

17. Hi Martin

If you are using plain MATLAB then it will be because MATLAB is using hardware-based double precision arithmetic and such a huge number cannot be exactly represented in it.

What you need to do is use the symbolic toolbox. Launch the mupad interface by typing

at the MATLAB prompt. Then, in the Mupad notebook, evaluate

isprime(N)

Replacing N with whatever huge integer you want to check. Mupad uses a very similar prime checking algorithm to Mathematica so the result is probabilistic (i.e., there is a chance it’s wrong!) as mentioned in a comment above.

Cheers,
Mike

18. @Mike Yeah… full notation is better since it gives the impression of “longness”. But anyway, a new notation.

Define a number in the form abbbb…bb where “a” and “b” are blocks of integers, and there are “n” blocks “b”‘s.

For a=7 and b=27, the numbers are prime for the following n’s:
n=1, 2, 4, 8, 35, 49, 121
For a=3 and b=73, the numbers are prime for the following n’s:
n=1, 10, 13, 40, 157, 424
For a=7 and b=87, the numbers are prime for the following n’s:
n=1, 2, 10, 13, 47

For a=1 and b=23456789, the numbers are prime for the following n’s:
n=59
For a=1 and b=87654321, the numbers are prime for the following n’s:
n=6, 49, 138

Yay! For more primes use the function:

19. Hi Mike,

thanks for the suggestion, that “mupad” command is quite useful for me!

Martin

20. Here is a quick brute-force search of two digit repeaters…

RepeatingDigits[n_, c_] :=
ToExpression[
Apply[StringJoin, Table[ToString[n], {c}]]
StringTake[ToString[n], 1]]
Do[If[PrimeQ[#], Print[#]] &[RepeatingDigits[n, c]], {n, 10, 99}, {c,
20, 80}]

1212121212121212121212121212121212121212121

1212121212121212121212121212121212121212121212121212121212121212121212121212121212121212121212121212121212121212121212121212121212121212121

151515151515151515151515151515151515151515151515151515151515151

15151515151515151515151515151515151515151515151515151515151515151515151515151515151515151

1616161616161616161616161616161616161616161616161616161

1616161616161616161616161616161616161616161616161616161616161616161616161616161616161616161616161616161616161

1616161616161616161616161616161616161616161616161616161616161616161616161616161616161616161616161616161616161616161616161616161616161616161616161

18181818181818181818181818181818181818181818181818181818181818181818181818181

1919191919191919191919191919191919191919191919191919191919191919191919191919191919191919191919191919191919191919191919191919191919191

313131313131313131313131313131313131313131313131313

31313131313131313131313131313131313131313131313131313131313131313131313131313131313

373737373737373737373737373737373737373737373737373737373737373737373737373737373

383838383838383838383838383838383838383838383838383838383

72727272727272727272727272727272727272727272727272727272727272727272727

727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727

75757575757575757575757575757575757575757575757575757575757575757575757575757

75757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757

75757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757575757

78787878787878787878787878787878787878787878787878787878787878787878787878787878787878787878787

94949494949494949494949494949494949494949494949494949494949494949

94949494949494949494949494949494949494949494949494949494949494949494949494949494949494949494949494949494949494949494949494949494949494949494949

979797979797979797979797979797979797979797979

98989898989898989898989898989898989898989898989898989898989898989898989898989898989898989898989898989898989898989898989898989898989898989898989898989898989898989

21. Oops. The blog comment system stripped out the StringJoin characters. Code again in different format…

RepeatingDigits[n_, c_] := ToExpression[ StringJoin[Apply[StringJoin , Table[ToString[n], {c}]], StringTake[ToString[n], 1]]];
Do[If[PrimeQ[#], Print[#]] & [RepeatingDigits[n, c]], {n, 10,
99}, {c, 20, 80}]

22. Another nice series are the “decimal repunit primes”, like 11, 1111111111111111111, and 11111111111111111111111. There are more: http://en.wikipedia.org/wiki/Repunit

23. 7272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727­27272727 is ‘nt prime
7272727272727272727272727272727272727272727272727272727272727272727272727272727272727272727­27272727 = 7*1153*1823071297393296313794645482344582929229459 *4942723325027431973353637942171511505058571593
PARI find it si not prime in 0ms and find the factorisation in 52 mn

24. Hi Paul

I copied your factorisation to Mathematica and got:
In[3]:= 7*1153*1823071297393296313794645482344582929229459*
4942723325027431973353637942171511505058571593

Out[3]= 72727272727272727272727272727272727272727272727272727272727272\
727272727272727272727272727277

The last digit is a 7 and not a 2