Problems & Puzzles: Puzzles

 Puzzle 1106 pkcp mod (skcp) = p JM Bergot sent the following nice puzzle:17*19*23*29 mod (17+19+23+29)=17 is an example of pkcp mod (skcp) = p where: pkcp means "Product of k consecutive primes" skcp means " Sum of k consecutive primes" p is the starting prime in pkcp and skcp Q. Can you find a longer list of consecutive primes that produces this?

During the week 9-15 Oct, 2022, contributions came from Michael Branicky, Jeff Heleen, Giorgos Kalogeropoulos, Adam Stinchcombe, Gennady Gusev, Paul Cleary, J-M Rebert, Oscar Volpati

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Michael wrote:

Length 5: ( 23 * ... * 41) mod ( 23 + ... + 41) = 23 Length 6: ( 1097 * ... * 1129) mod ( 1097 + ... + 1129) = 1097 Length 6: (18313 * ... * 18371) mod (18313 + ... + 18371) = 18313 Length 10: ( 59 * ... * 101) mod ( 59 + ... + 101) = 59 Length 20: ( 73 * ... * 173) mod ( 73 + ... + 173) = 73 Length 44: ( 4133 * ... * 4513) mod ( 4133 + ... + 4513) = 4133 Length 46: ( 4111 * ... * 4507) mod ( 4111 + ... + 4507) = 4111 Length 62: ( 4157 * ... * 4679) mod ( 4157 + ... + 4679) = 4157 Length 923: ( 4027 * ... * 12391) mod ( 4027 + ... + 12391) = 4027

Length 170: (36293 * ... * 38047) mod (36293 + ... + 38047) = 36293 Length 251: (33629 * ... * 36313) mod (33629 + ... + 36313) = 33629

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Jeff wrote:

For starting primes less than 5000 I have found 7 other solutions:

23 to 41 (5 primes)
59 to 101 (10 primes)
73 to 173 (20 primes)
1097 to 1129 (6 primes)
4111 to 4507 (46 primes)
4133 to 4513 (44 primes)
4157 to 4679 (62 primes)

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Giorgos wrote:

Here is a longer list with 11924 consecutive primes starting from p = 8069.
8069 * 8081 * 8087 * 8089 * . . . * 139177 mod (8069 + 8081 + 8087 + 8089 + . . . + 139177) = 8069

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I got 13 p's with length longer than 4, the longer ones being p=4027 of length 923 and 45503 of length 970 and the largest p (so far) being 905497 of length 33.

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The largest result is 11924 consecutive primes from 8069.
I have found several examples a longer list than in puzzle statement:
n, first prime
5, 23

6, 1097

6, 18313

10, 59

20, 73

33, 905497

44, 4133

46, 4111

62, 4157

170, 36293

251, 33629

848, 462131

923, 4027

970, 45503

1631, 173543

2415, 235891

3116, 361961

4009, 534511

5278, 141371

9494, 264619

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Paul wrote:

Here are a few more examples with longer lists.

No of consec primes =4 , 17 *..29 Mod 17 +..29 = 17

No of consec primes =5 , 23 *..41 Mod 23 +..41 = 23

No of consec primes =6 , 1097 *..1129 Mod 1097 +..1129 = 1097

No of consec primes =6 , 18313 *..18371 Mod 18313 +..18371 = 18313

No of consec primes =10 , 59 *..101 Mod 59 +..101 = 59

No of consec primes =20 , 73 *..173 Mod 73 +..173 = 73

No of consec primes =33 , 905497 *..905959 Mod 905497 +..905959 = 905497

No of consec primes =44 , 4133 *..4513 Mod 4133 +..4513 = 4133

No of consec primes =46 , 4111 *..4507 Mod 4111 +..4507 = 4111

No of consec primes =62 , 4157 *..4679 Mod 4157 +..4679 = 4157

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Jean-Marc wrote:

k p
4 17
5 23
6 1097
10 59
20 73
33 905497
44 4133
46 4111
62 4157
170 36293
251 33629
848 462131
923 4027
970 45503
1631 173543
2415 235891
3116 361961
4009 534511
5278 141371

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Oscar wrote:

Among the first 20000 primes, I found several solutions, with lengths between 5 and 11924.

k  p  q  skcp
4  17  29  88
5  23  41  161
6  1097  1129  6678

10  59  101  780
20  73  173  2448
44  4133  4513  190118

46  4111  4507  197972
62  4157  4679  274362

170  36293  38047  6314982

251  33629  36313  8777169
923  4027  12391  7496979

970  45503  56041  49257230
1631  173543  193073  298990013

5278  141371  205297  914033740
11924  8069  139177  850504876

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