Problems & Puzzles: Puzzles

Puzzle 1118 Prime[n] + k*(Prime[n + 1] - Prime[n]) Sebastián Martín Ruiz sent the following nice puzzle: I have observed that Prime[n] + 6*(Prime[n + 1] - Prime[n]) is prime from n=3 to 18, producing the following 16 prime values: 17,31,23,37,29,43,59,41,67,61,53,67,83,89,71,97 Q. Find a larger series of successive primes for Prime[n] + k*(Prime[n + 1] - Prime[n]) and another k>0 fixed integer value.

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During the week 7-13 Jan, 2023 contributions came from J-M Rebert, Mauro Fiorentini, Giorgio Kalogeropoulos, Emmanuel Vantieghem, Gennady Gusev, Oscar Volpatti, Martin Hopf

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Jean-Marc wrote:

Q. The longest series of successive primes I found is a infinite series for k=1. prime(n)+ k*(prime(n+1)-prime(n)) =prime(n+1) is prime for all n >= 1 .

For k > 1, I found the following series for n = n0 to n0 + length -1:

length, k, n0

4 2 3483
8 3 279
16 6 3
16 76 64
16 6511 10811
17 6916 67068
17 7561 2370
17 7645 97878
19 15225 17842

 

solution(2,3483)                                            

f(n,k)=Prime[n] + k*(Prime[n + 1] - Prime[n]) is prime from n = 3483 to 3486, the following primes : 

[32491, 32491, 32503, 32503]

, sorted and without duplicate, the list [32491, 32503] has only 2 primes.

 

 solution(3,279)

f(n,k)=Prime[n] + k*(Prime[n + 1] - Prime[n]) is prime from n = 279 to 286, the following primes : 

[1831, 1847, 1847, 1879, 1889, 1879, 1879, 1877]

, sorted and without duplicate, the list [1831, 1847, 1877, 1879, 1889] has only 5 primes.

 

solution(6,3)

f(n,k)=Prime[n] + k*(Prime[n + 1] - Prime[n]) is prime from n = 3 to 18, the following primes : 

[17, 31, 23, 37, 29, 43, 59, 41, 67, 61, 53, 67, 83, 89, 71, 97]

, sorted and without duplicate, the list [17, 23, 29, 31, 37, 41, 43, 53, 59, 61, 67, 71, 83, 89, 97] has only 15 primes.

 

solution(76,64)

f(n,k)=Prime[n] + k*(Prime[n + 1] - Prime[n]) is prime from n = 64 to 79, the following primes : 

[463, 617, 1381, 787, 1097, 499, 653, 809, 967, 823, 829, 683, 839, 997, 701, 1009]

, sorted and without duplicate, the list [463, 499, 617, 653, 683, 701, 787, 809, 823, 829, 839, 967, 997, 1009, 1097, 1381] has also 16 primes.

 

 solution(6511,10811)

f(n,k)=Prime[n] + k*(Prime[n + 1] - Prime[n]) is prime from n = 10811 to 10826, the following primes : 

[309559, 179369, 166357, 140321, 231479, 192431, 166399, 179429, 205483, 296651, 153437, 309707, 192539, 322771, 218627, 153533]

, sorted and without duplicate, the list [140321, 153437, 153533, 166357, 166399, 179369, 179429, 192431, 192539, 205483, 218627, 231479, 296651, 309559, 309707, 322771] has also 16 primes.

 

solution(6916,67068)

f(n,k)=Prime[n] + k*(Prime[n + 1] - Prime[n]) is prime from n = 67068 to 67084, the following primes : 

[856249, 870083, 1008407, 856279, 1008433, 870137, 883973, 883979, 897817, 911657, 925499, 856351, 911681, 980851, 1050031, 884077, 925579]

, sorted and without duplicate, the list [856249, 856279, 856351, 870083, 870137, 883973, 883979, 884077, 897817, 911657, 911681, 925499, 925579, 980851, 1008407, 1008433, 1050031] has also 17 primes.

 

 solution(7561,2370)

f(n,k)=Prime[n] + k*(Prime[n + 1] - Prime[n]) is prime from n = 2370 to 2386, the following primes : 

[187409, 111821, 66467, 126961, 157219, 51383, 66509, 81637, 66523, 66529, 96779, 81667, 51431, 36313, 157291, 96821, 66587]

, sorted and without duplicate, the list [36313, 51383, 51431, 66467, 66509, 66523, 66529, 66587, 81637, 81667, 96779, 96821, 111821, 126961, 157219, 157291, 187409] has also 17 primes.

 

 solution(7645,97878)                                        

f(n,k)=Prime[n] + k*(Prime[n + 1] - Prime[n]) is prime from n = 97878 to 97894, the following primes : 

[1575401, 1284931, 1575443, 1330843, 1361431, 1483763, 1285021, 1346183, 1376773, 1575557, 1652047, 1361587, 1346309, 1285159, 1499221, 1315771, 1300487]

, sorted and without duplicate, the list [1284931, 1285021, 1285159, 1300487, 1315771, 1330843, 1346183, 1346309, 1361431, 1361587, 1376773, 1483763, 1499221, 1575401, 1575443, 1575557, 1652047] has also 17 primes.

 

 solution(15225,17842)

f(n,k)=Prime[n] + k*(Prime[n + 1] - Prime[n]) is prime from n = 17842 to 17860, the following primes : 

[228707, 472309, 259177, 502781, 381001, 350563, 411473, 350587, 228797, 624649, 411527, 289741, 381097, 259309, 411563, 350677, 228887, 533389, 228911]

, sorted and without duplicate, the list [228707, 228797, 228887, 228911, 259177, 259309, 289741, 350563, 350587, 350677, 381001, 381097, 411473, 411527, 411563, 472309, 502781, 533389, 624649] has also 19 primes.

 

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Mauro wrote:

Another sequence of the same length occurs for k = 76 and n from 64 to
79, producing the following 16 prime values: 463, 617, 1381, 787, 1097,
499, 653, 809, 967, 823, 829, 683, 839, 997, 701, 1009.

There are also 3 sequences with length 15:
for k = 6 and n from 4 to 18, producing the following 15 prime values:
31, 23, 37, 29, 43, 59, 41, 67, 61, 53, 67, 83, 89, 71, 97;
for k = 76 and n from 65 to 79, producing the following 15 prime
values: 617, 1381, 787, 1097, 499, 653, 809, 967, 823, 829, 683, 839,
997, 701, 1009;
for k = 9346 and n from 710 to 724, producing the following 15 prime
values: 5387: 61463, 61469, 80167, 61483, 42797,
24109, 117571, 61507, 42821, 24133, 61519, 211061, 61547, 24169, 42863.

No other sequence longer than 14 terms for k up to 10000 and n up to
1000.

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Giorgio wrote:

Prime[n] + 15225*(Prime[n + 1] - Prime[n])
is prime from n=17842 to 17860, producing the following 19 prime values: 228707, 472309, 259177, 502781, 381001, 350563, 411473, 350587, 228797, 624649, 411527, 289741, 381097, 259309, 411563, 350677, 228887, 533389, 228911

 

Here are also some other values for k and n that produce 19 primes:

       k      |       n

75810   -> 9981

107296 -> 722758

326886 -> 84932

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Emmanuel wrote:

The expression
     p(n) + 7561*(p(n+1) - p(n)
is prime from  n = 2370  to  2386, producing the following 17 prime values :
   187409, 111821, 66467, 126961, 157219, 51383, 66509, 81637, 66523, 66529, 96779, 81667, 51431, 36313, 157291, 96821, 66587
The expression
     p(n) + 15225*(p(n+1) - p(n)
is prime from  n = 17842 to 17860, producing the following 19 prime values :
   228707, 472309, 259177, 502781, 381001, 350563, 411473, 350587, 228797, 624649, 411527, 289741, 381097, 259309, 411563, 350677, 228887, 533389, 228911

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Gennady wrote:

A larger series of successive primes for different k <100000 and 10^8 primes:

17: k=6916, n=67068, p(n)=842417

19: k=15225, n=17842, p(n)=198257

18: k=36960, n=35465296, p(n)=683976427

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Oscar wrote:

I found solutions with length from 17 to 19.

 

Solution with length 17.
k = 7561
n = 2370 to 2386
p(n) = 21067 to 21221
produced primes:
187409, 111821, 66467, 126961, 157219, 51383, 66509, 81637, 66523, 66529, 96779, 81667, 51431, 36313, 157291, 96821, 66587.

 

Solution with length 19.
k = 116655
n = 7581 to 7599
p(n) = 77167 to 77347
produced primes:
543787, 2410271, 1243741, 1477061, 2876933, 310547, 543859, 777173, 1477109, 310571, 543883, 310577, 1243819, 1477139, 3110321, 777247, 1943803, 1010579, 543967.

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Martin Hopf

I found the following 17-series:
p = prime(n) + k*(prime(n+1) - prime(n))

k = 6916; n = [67068..67084]
p = [856249, 870083, 1008407, 856279, 1008433, 870137, 883973, 883979, 897817, 911657, 925499, 856351, 911681, 980851, 1050031, 884077, 925579]

k = 7561; n = [2370..2386]
p = [187409, 111821, 66467, 126961, 157219, 51383, 66509, 81637, 66523, 66529, 96779, 81667, 51431, 36313, 157291, 96821, 66587]

k = 7645; n = [97878..97894]
p = [1575401, 1284931, 1575443, 1330843, 1361431, 1483763, 1285021, 1346183, 1376773, 1575557, 1652047, 1361587, 1346309, 1285159, 1499221, 1315771, 1300487]

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