Problems & Puzzles: Puzzles

Puzzle 40.- The Pi Prime Search Puzzle (by Patrick De Geest)

PI with 301 digits, according to the program PI ub

3.14159265358979323846264338327950288419716939937510582
0974944592307816406286208998628034825342117067982148086
5132823066470938446095505822317253594081284811174502841
0270193852110555964462294895493038196442881097566593344
6128475648233786783165271201909145648566923460348610454
3266482133936072602491412737

The following Pi-Search Page inspired me to construct this puzzle :

http://www.aros.net/~angio/pi_stuff/piquery.html

As Pi is one of the most known mathematical constants, I'd like to present the puzzler the following prime search assignments.

PART A

Question :

Find ever larger prime strings in the decimal expansion of Pi starting at a location that is also prime (location starts at the first position right after the decimal point, the 3. is not counted). Furthermore the prime found at that prime location must also be of that 'prime location' length.

The first easy solution is of course :

Prime Location Number string

---------------------------------
2, 41 [ 41 is prime ]

In this case we could say that at 'prime location 2' we found a prime string of 'length 2' !

3, 159 [ 3 x 53 ]

At prime location 3 we find the string '159'. Not a good solution as 159 is composite. So, let's move on to the next candidate :

5 92653 [ 11 x 8423 ]

Again, the numberstring is composite.

7 6535897 [ bingo! ]

And yes, our second prime solution enters the puzzlegame.

The last number I checked was at position 19 : 4626433832795028841 but alas again composite but only just as it has two factors : 61396271 x 75353661671.

Now, all we need to ask the puzzler is to extend this list.
Maybe he likes to find the next one or maybe he prefers to find directly a very large solution. It's the puzzler's choice!

 Prime location Number string of length ‘prime location’ that is also a prime number. 2 41 7 6535897 ¿ ¿

Felice Russo, from Italy, at 23/03/99 wrote: "I didn't find any other prime number up to position 151".

PART B

Question :

In the decimal portion of "Pi", find the first position where appears a
prime of length 1,2,3,4,5....

One can look for prime of length say 3  (any one from 101 to 997)
A quick glimpse at the expansion reveals that 653 is ‘the winner’ at location 7.

Searching for ‘the loser’ (in this case the last 3 digits prime to appear) might be equally interesting! For length 3 this turns out to be a 'palindromic prime' namely 373 !… at position 5229!!!

 Length prime Winner prime & Position Loser prime & Position 1 5 4 7 13 2 41 2 73 299 3 653 7 373 5229 =>4 4,  4159 (position 2) 5,  14159 (position 1) 6,   358979 (position 9) 7,   1592653 (position 3) 8,  28841971 (position 33) (sent by Jim Howell, at 23/02/99) 4, 9337 (position 75 961) 5, 35569 (position 715492) (sent by J. C. Colin, October 29, 2002)

Can you extend this table?

I'll list just a handful here to get you on the way :

Science : Mathematics : Numerical Analysis : Numbers :
Specific Numbers : Pi : Calculating Pi.

Did you know : Ubasic has a 'PI.UB' file that calculates up to 2500 digits.

J. C. Colin sent (October 29, 2002) the following results for the winner and looser primes in Pi, asked by Patrick:

 Length prime Winner prime position 9 795028841 29 10 5926535897 4 11 93238462643 14 12 141592653589 1 13 9265358979323 5 14 23846264338327 16 15 841971693993751 35 16 8628034825342117 81 17 89793238462643383 11 18 348253421170679821 86 19 3832795028841971693 25 20 89793238462643383279 11 21 338327950288419716939 24 22 9334461284756482337867 214 23 88419716939937510582097 34 24 384626433832795028841971 17 25 2384626433832795028841971 16 26 41592653589793238462643383 2 27 169399375105820974944592307 40 28 2384626433832795028841971693 16 29 86783165271201909145648566923 233 30 238462643383279502884197169399 16 31 7019385211055596446229489549303 166 32 34211706798214808651328230664709 91 33 145648566923460348610454326648213 250 34 9323846264338327950288419716939937 14 35 53589793238462643383279502884197169 8 36 897932384626433832795028841971693993 11 37 9502884197169399375105820974944592307 30 38 41592653589793238462643383279502884197 2 39 749445923078164062862089986280348253421 56 40 2602491412737245870066063155881748815209 289 41 15926535897932384626433832795028841971693 3 42 679821480865132823066470938446095505822317 98 43 4461284756482337867831652712019091456485669 217 44 98336733624406566430860213949463952247371907 501 45 751058209749445923078164062862089986280348253 47 46 1027019385211055596446229489549303819644288109 163 47 81964428810975665933446128475648233786783165271 197 48 644288109756659334461284756482337867831652712019 200 49 6095505822317253594081284811174502841027019385211 127 50 26535897932384626433832795028841971693993751058209 6 51 113305305488204665213841469519415116094330572703657 362 52 3594081284811174502841027019385211055596446229489549 142 53 58979323846264338327950288419716939937510582097494459 10 54 317253594081284811174502841027019385211055596446229489 137 55 7938183011949129833673362440656643086021394946395224737 486 56 50288419716939937510582097494459230781640628620899862803 31 57 233786783165271201909145648566923460348610454326648213393 229 58 8628034825342117067982148086513282306647093844609550582231 81 59 25903600113305305488204665213841469519415116094330572703657 354 60 656643086021394946395224737190702179860943702770539217176293 514 61 2829254091715364367892590360011330530548820466521384146951941 333 62 22948954930381964428810975665933446128475648233786783165271201 185 63 105559644622948954930381964428810975665933446128475648233786783 175 64 8475648233786783165271201909145648566923460348610454326648213393 222 65 74944592307816406286208998628034825342117067982148086513282306647 56 66 822317253594081284811174502841027019385211055596446229489549303819 134 67 8841971693993751058209749445923078164062862089986280348253421170679 34 68 74944592307816406286208998628034825342117067982148086513282306647093 56 69 867831652712019091456485669234603486104543266482133936072602491412737 233 70 1748815209209628292540917153643678925903600113305305488204665213841469 319

As a matter of fact he has extended this table up to length=104; he has also produced one winner prime whose length = 1499, and begins at the position 169. He verified the primality with PRIMO (2 day! with a 435MHz Pentium II).

***

Other result sent by J. C. Colin are:

I think the loser prime of length 6 is :
805 289 which begins at the position 11 137 824 in the decimal digits of pi.

***

The loser prime of length = 7 is 9 271 903 at the position 135 224 164 in the decimal digits of pi.
The method I use to find this result is :
1- I made 9 files contained all the primes with 7 digits

 file first  prim last prime tot 1 p(78499) = 1M +3 p(148933) = 2M -7 70435 2 p(148934) = 2M +3 p(216816) = 3M -1 67883 3 p(216817) = 3M +17 p(283146) = 4M -29 66330 4 p(283147) = 4M +37 p(348513) = 5M -1 65367 5 p(348514) = 5M +11 p(412849) = 6M -7 64336 6 p(412850) = 6M +11 p(476648) = 7M -3 63799 7 p(476649) = 7M +3 p(539777) = 8M -7 63129 8 p(539778) = 8M +9 p(602489) = 9M -7 62712 9 p(602490) = 9M +11 p(664579) = 10M -9 62090
Total =  586 081
2- for each file, I made 107 numbers b(i) resulting of the multiplication of the prime. I search in the first 20 millions decimal digits of pi if number of length =7 are prime and in b(i).
At the end of this calculation , I find 506 052 primes/ 586081 primes of length = 7. So it remains 80 029 primes not find.

3- with its 80029 primes, I calculate 131 b(i) and I search in the successive millions decimal digits of pi if they are presents. Then I find
up to about 30 millions ( 29 999 673 exactly) it remains 29 530 primes
up to about 40 millions, it remains 10 738
up to about 60 millions, it remains 1 495 primes of length=7 not find in  the decimal digits of pi
up to about 80 millions, it remains 218 primes
up to about 100 millions, it remains 33 primes
up to about 120 millions, it remains  3 primes = 1 111 723, 3 557 327 and 9 271 093
and finally up to about 130 millions , only 9 271 093 stays.
With the next downloaded file which contains 10 millions of decimal digits of pi for 130 millions up to 140 millions, I found the position  of  9 271 093 : 135 224 164.

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