Multiple palindrome primes (PRPUs) with the same
sum over all integers, same number of digits and the
smallest number of non zero integers (4).
Palprimes involving one or two non zero integers of the form p0I0p or
p0I0q0I0p cannot possess multiple primes with the same number of
integers and the same sum over all integers with the possible exception
of 10^k + 1 where 11 and 101 are the only primes found so far. The simplest
example of palprimes with these properties are numbers of the form
N(p,q,k,m) = p.10(2k) + q.10(k+m) + q.10(k-m) + p
where m can take on all values from m = 1 to m = k – 1. This palindrome
number has the properties that its total number of digits is 2k+1 and the
sum over all digits is 2(p+q).
For numbers of this type the largest has the largest value of k. If two or
more numbers have the same value of k the largest number is the one with
the largest value of p. If numbers have the same value of both k and p the
size is determined by the largest value of m. In case all three constants are
the same the final size arbiter is the size of q.
In order to be prime N must have an odd number of digits, which it does, p
cannot be an even integer or 5, p + q cannot be divisible by 3, and p cannot
be equal to q. Earlier work suggests that p = q = 1 palindrome numbers
cannot be prime. This leaves 22 possible values of combinations of p and q
for the production of palprimes as shown in Table 1 where primes with fixed
p, q, and k with the largest values of k < 1501 and the most number of mUs
are given. Also given in Table 1 are the total number of primes found in a
search from k = 2 to 751 and a search from k = 752 to 1501 (3003 digits).
The total number of primes found for fixed values of p and q and a
particular range of k values is roughly constant (about 10% standard
deviation) as shown in Table 2. If the number of primes found is A for a
search over a range of k values dk where the average k value for the
range is k(ave) then the prime density, D, can be shown to be
D = A/{dk [ k(ave)+0.5]}
Hence the prime density falls off linearly with the average value of k for a
series of searches of constant width dk.
It is possible to have multiple primes with the same total number of digits
and sum over all digits. This corresponds to fixed values of p, q, and k and
a prime search over values of m from 1 to k-1. There are 8 different sets of
combinations of p and q as shown in Table 3. A second property of possible interest
is to fix the sum of p + q and value of k and look for the primes N(p,q,k,m) and
N(q,p,k,m) with the same value of m. Note that in the previous case p and q are not
always the same pair of integers. These two primes are identical in all respects
except that p and q in one prime becomes q and p in the other prime. Of course the
larger of the two primes is the one with the larger value of p. Table 4 shows the
possible five groups for k < 1501 where the sum of p and q is the same.
Table 1: Largest palprimes with k < 1501 with the largest number
of m values and the same values of k, p, and q.
Number of palprimes between k = 2 and 751 and between 752 and 1501
are displayed in the last two columns.
| case | p | q | k | m | # of primes
k = 2-751 | # of primes
k= 752-1501 |
|---|
| 1 | 1 | 3 | 1154 | 119,132,223,775,827,1087 | 507 | 509 |
| 2 | 1 | 4 | 1095 | 420,502,670,952,956 | 609 | 595 |
| 3 | 1 | 6 | 848 | 132,167,695,713,732 | 450 | 456 |
| 4 | 1 | 7 | 1408 | 35,314,483,610,929 | 619 | 641 |
| 5 | 1 | 9 | 730 | 79,124,289,447,469 | 634 | 574 |
| 6 | 3 | 1 | 808 | 15,48,113,603,640,783 | 555 | 550 |
| 7 | 3 | 2 | 1488 | 137,492,920,1346 | 583 | 572 |
| 8 | 3 | 4 | 984 | 95,123,139,262,758 | 598 | 529 |
| 9 | 3 | 5 | 760 | 31,68,133,272,334,508,575,650,658 | 534 | 515 |
| 10 | 3 | 7 | 1184 | 93,225,396,659,722 | 478 | 508 |
| 11 | 3 | 8 | 1357 | 158,718,1042,1210 | 383 | 366 |
| 12 | 7 | 1 | 1282 | 553,832,1151,1183,1223 | 619 | 652 |
| 13 | 7 | 3 | 1362 | 257,455,611,840,905,1236 | 521 | 543 |
| 14 | 7 | 4 | 1396 | 523,751,841,1107,1281 | 422 | 442 |
| 15 | 7 | 6 | 1452 | 494,623.634,811,1090,1437 | 419 | 468 |
| 16 | 7 | 9 | 472 | 34,45,92,327,370,446 | 625 | 625 |
| 17 | 9 | 1 | 1350 | 135,245,437,1180, 1231,1340 | 549 | 617 |
| 18 | 9 | 2 | 780 | 67,149,503 | 105 | 75 |
| 19 | 9 | 4 | 1493 | 610,1024,1316,1330 | 490 | 463 |
| 20 | 9 | 5 | 1277 | 576.864.1130,1210 | 419 | 423 |
| 21 | 9 | 7 | 1090 | 235,373,680,809,868 | 632 | 589 |
| 22 | 9 | 8 | 1280 | 482,775,1137,1154,1208 | 491 | 508 |
Table 2: Number of primes found in k intervals
of width 100 for searches from k = 1000 to 2000
for p = 7 and q = 9.
| k = 1000 → 1100 | 74 |
| k = 1100 → 1200 | 67 |
| k = 1200 → 1300 | 84 |
| k = 1300 → 1400 | 83 |
| k = 1400 → 1500 | 78 |
| Average +/- std. dev. | 77 +/- 7 |
| k = 1500 → 1600 | 93 |
| k = 1600 → 1700 | 74 |
| k = 1700 → 1800 | 87 |
| k = 1800 → 1900 | 72 |
| k = 1900 → 2000 | 81 |
| Average +/- std. dev. | 81 +/- 9 |
Table 3. Largest palindrome primes with fewer than 1500 digits
with the same sum over all digits and the same number of digits
| k | p | q | m | Total
# digits | Sum of
digits |
|---|
| 655 | 1 | 3 | 64,342,371,458 | 1313 | 8 |
| | 3 | 1 | 334,408,434 | | |
| 515 | 1 | 4 | 105,217,232,299 | 1033 | 10 |
| | 3 | 2 | 17,307,443 | | |
| 323 | 1 | 6 | 57,103,323 | 649 | 14 |
| | 3 | 4 | 152,193,272,286 | | |
| 124 | 7 | 6 | 4,50 | 251 | 26 |
| | 9 | 4 | 14,96,118 | | |
| 471 | 7 | 9 | 34,45,92,327,370,446 | 945 | 32 |
| | 9 | 7 | 77,265,286,294,425 | | |
| 687 | 1 | 7 | 34,61 | 1377 | 16 |
| | 3 | 5 | 34,200 | | |
| | 7 | 1 | 218,373,407,501,594 | | |
| 498 | 3 | 8 | 232,276,396 | 999 | 22 |
| | 7 | 4 | 146,236,326 | | |
| | 9 | 2 | None | | |
| 453 | 1 | 9 | 49,340,440 | 909 | 20 |
| | 3 | 7 | 108 | | |
| | 7 | 3 | 252,365,371,412,424 | | |
| | 9 | 1 | 193,401 | | |
Table 4: Palprime pairs N(p,q,k,m) and N(q,p,k,m).
The largest palprimes found are shown for all pairs k < 1501
and the total number found in each category except for the case
of 1 - 3 and 3 - 1 where the three largest primes are also listed.
| Range of k | p(1) | q(1) | p(2) | q(2) | k | m | Number of pairs found |
|---|
| 1-1500 | 1 | 3 | 3 | 1 | 412 | 263 | 7 |
| 1-1500 | 1 | 3 | 3 | 1 | 1156 | 688 | 7 |
| 1-1500 | 1 | 3 | 3 | 1 | 1232 | 366 | 7 |
| 1-1500 | 1 | 7 | 7 | 1 | 33 | 6 | 2 |
| 1-1500 | 1 | 9 | 9 | 1 | 629 | 622 | 4 |
| 1-1500 | 3 | 7 | 7 | 3 | 184 | 173 | 4 |
| 1-1500 | 7 | 9 | 9 | 7 | 52 | 29 | 1 |
|
Extra N(p,q,k,m) data research
Number N is prime for value(s) m
N(1,3,9,m) → m = 4
N(1,4,9,m) → m = 2
N(1,6,9,m) → none
N(1,7,9,m) → none
N(1,9,9,m) → none
N(3,1,9,m) → none
N(3,2,9,m) → none
N(3,4,9,m) → none
N(3,5,9,m) → m = 2
N(3,7,9,m) → none
N(3,8,9,m) → none
N(7,1,9,m) → m = 2, 8
N(7,3,9,m) → none
N(7,4,9,m) → m = 2
N(7,6,9,m) → none
N(7,9,9,m) → none
N(9,1,9,m) → none
N(9,2,9,m) → none
N(9,4,9,m) → m = 6
N(9,5,9,m) → m = 2, 8
N(9,7,9,m) → m = 4
N(9,8,9,m) → none
|
From the 22 * 8 or 176 candidates (with k = 9),
just 10 numbers turn out to be prime.
Can you find the smallest value k > 9 yielding no primes at all ?
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Multiple Palindromic Primes
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