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?

Sites about Pi are abundant.
I'll list just a handful here to get you on the way :

The best place to start with is Yahoo (www.yahoo.com )
via the following menu cascading :
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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