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^{(km)} + 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 k1. 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 = 2751  # of primes k= 7521501 
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 
11500  1  3  3  1  412  263  7 
11500  1  3  3  1  1156  688  7 
11500  1  3  3  1  1232  366  7 
11500  1  7  7  1  33  6  2 
11500  1  9  9  1  629  622  4 
11500  3  7  7  3  184  173  4 
11500  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 ?
