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.
***
