SPT
[[File:{{#setmainimage:Spt.jpg}}|alt=logo image|center|frameless]] SPT is a BOINC based volunteer computing project that needs your help to research Symmetric Prime Tuples.
Goal
To continue the work of the T. Brada Experimental Grid project.
Methods
Definition 1
A prime k-tuple is a finite collection of values (p + a1, p + a2, p + a3, …, p + ak),
where p, p + a1, p + a2, p + a3, …, p + ak are prime numbers, (a1, a2, a3, …, ak) are pattern. Typically the first value in the pattern is 0 and the rest are distinct positive even numbers. [1]
We consider the k-tuple, where p + a1, p + a2, p + a3, ..., p + ak are consecutive primes.
Definition 2
k-tuple (p + a1, p + a2, p + a3, ..., p + a [k / 2], p + a [k / 2+1], ..., p + a [k-2], p + a [k-1], p + ak) for k even, is called symmetric, if the following condition is satisfied:
a1 + ak = a2 + a[k-1] = a3 + a[k-2] = … = a[k/2] + a[k/2+1]
Example
symmetric 8-tuple
(17 + 0, 17 + 2, 17 + 6, 17 + 12, 17 + 14, 17 + 20, 17 + 24, 17 + 26)
Shortened we write this:
17: 0, 2, 6, 12, 14, 20, 24, 26
Definition 3
k-tuple (p + a1, p + a2, p + a3, ..., p + a [(k-1) / 2], p + a [(k-1) / 2 + 1], p + a [(k-1) / 2 + 2], ..., p + a [k-2], p + a [k-1], p + ak) for k odd called symmetric, if the following condition is satisfied:
a1 + ak = a2 + a[k-1] = a3 +a [k-2] =…= a[(k-1)/2] + a[(k-1)/2+2] = 2 a[(k-1)/2+1]
Example
symmetric 5-tuple
18713: 0, 6, 18, 30, 36
(See in [2])
Definition 4
The diameter d of k-tuple is the difference of its largest and smallest elements. [1]
Example
8-tuple
17: 0, 2, 6, 12, 14, 20, 24, 26
It has a diameter d = 26.
Known solutions with a minimal diameter and a minimal value of p for k = 2, 4, 6, 8 see in [1]
Further,
k=5 ( d, p minimal)
18713: 0, 6, 18, 30, 36
k=7 (d, p minimal)
12003179: 0, 12, 18, 30, 42, 48, 60
k=9 (d, p minimal)
1480028129: 0, 12, 24, 30, 42, 54, 60, 72, 84
k=10 (d, p minimal)
13: 0, 4, 6, 10, 16, 18, 24, 28, 30, 34
k=11 (possible minimal ?)
660287401247651: 0, 6, 30, 42, 60, 66, 72, 90, 102, 126, 132
k=12 (not minimal)
137: 0, 2, 12, 14, 20, 26, 30, 36, 42, 44, 54, 56
k=13 (not minimal)
5348080416833699: 0, 12, 30, 42, 48, 72, 90, 108, 132, 138, 150, 168, 180
k=14 (not minimal)
19636011281690651: 0, 2, 8, 12, 18, 26, 30, 38, 42, 50, 56, 60, 66, 68
k=15 (not minimal)
5348080416833681: 0, 18, 30, 48, 60, 66, 90, 108, 126, 150, 156, 168, 186, 198, 216
k=16 (not minimal)
19636011281690647: 0, 4, 6, 12, 16, 22, 30, 34, 42, 46, 54, 60, 64, 70, 72, 76
k=18 (not minimal)
49549273441123: 0, 4, 24, 40, 46, 54, 58, 66, 70, 84, 88, 96, 100, 108, 114, 130, 150, 154
k=20 (not minimal)
11785542108641839: 0, 4, 10, 18, 24, 30, 52, 70, 72, 84, 118, 130, 132, 150, 172, 178, 184, 192, 198, 202
k=22 (not minimal)
18620445306703861: 0, 10, 36, 46, 66, 76, 82, 96, 102, 130, 136, 162, 168, 196, 202, 216, 222, 232, 252, 262, 288, 298
k=24 (not minimal)
22930603692243271: 0, 70, 76, 118, 136, 156, 160, 178, 202, 222, 238, 250, 378, 390, 406, 426, 450, 468, 472, 492, 510, 552, 558, 628
(See [3])
For k = 17, 19, 21, 23 solutions no found.
Questions
1. Find solutions with a minimal diameter and a minimal value of p for 10 < k < 17, k = 18, 20, 22, 24.
2. Find solutions for the remaining k, minimal or not minimal.
Project team
Natalia Makarova. Alex Belyshev. Tomáš Brada.
Scientific results
Results of this project are available in the Prime Tuple Database.