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Palindromic Primes where the sum
over all digits is a minimum
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Introduction

What is known about palindrome primes of the form

N(n,k) = 102n + 10n + 1 + 10(2nk) + 10k ; 0 < k < n
and
N(n) = 102n + 3*10n + 1  ?

The sum over all digits in these primes is always 5 which except for
single digit primes (2, 3) and 11 and 101 would seem to be the smallest possible.
These palprimes might have the highest percentage of zeros and involve the smallest
number of different digits. The second type, N(n) has a maximum of one prime
with a fixed number of digits while the first type, N(n,k), can have more than one prime
with the same number of digits. The largest prime of the first type I have found is a
meager 5000 digits with the greatest number of primes with the same number of digits equal to 5.
The number of primes of the first type found in a constant search area of n [n(max) – n(min) = 200]
is nearly a constant value (~137). The 19th prime of type 2 is a little over 11000 digits.
I think I have a partial proof (conjecture?) that there are no primes with a sum over digits of 2
beyond 101 and proofs that palindromes with a sum over all digits of 3 and 4 do not contain primes.

Email Russell Bonham


Article by Russell Bonham

Palprimes where the sum over all digits is a minimum

The minimum sum over all digits of palindrome primes is 2,
excluding single digit primes, which occurs for the primes 11 and
101. It is conjectured that there are no other palprimes with a sum
over all digits of 2 and it is proved that all palindrome numbers
with a sum over all digits of 3 and 4 are not prime. The next
smallest sum over all digits is 5 for which there are two cases
which exhibit primes and are explored below.

The palindromes N2[m] = 10m + 1 have no primes for m > 2.

Conjecture and partial proof: Since
2n   
x2n+1 + 1 = (1 + x){S (–1) px p} 
 p = 0 
it can be concluded that N2[2m + 1] = 102m+1 + 1 contains no primes for m > 0.
For even values of m, N2[2m], can be written as N2[k,n] = 102k(2n+1) + 1
where by the argument above N2[k,n] contains no primes for n > 0
which leaves the n = 0 case, D[k] = 102k + 1 as the only possible
numbers which might be prime. Note that D(k) is the greatest
common denominator (GCD) of N2[k,n] for n > 0.

The lack of primality of D(k) was tested by calculating the smallest
denominator (SD) of D(k) by use of "Mathematica's" IntegerQ[n] function
on the ratio D(k)/Prime[m] with m increasing from 1 to until IntegerQ[ratio]
was True. The result was checked by dividing SD into D(k) and using IntegerQ[n]
on the result. The results for the SD's are displayed in Table 1. This means
that the smallest possible prime for D(k) with k > 1 must have at least 2097152
digits (k = 21).

Palindromes with a sum over all digits equal to 3 are not prime.

Proof: The palindrome numbers N3[m] = 102m + 10m + 1 have a GCD of 3
for m = 1 through 3. Assuming that N3[m] is divisible by 3 then it
can be shown that N3[m + 1]/3 = {100*N3[m] – 9*10m+1 – 99}/3 so that there
are no palindrome primes with a sum over all digits of 3.

Palindromes with a sum over all digits equal to 4 are not prime.

Proof: Palindrome numbers with a sum over all digits of 4 can be
written as N4[n,m] = 10m+n + 10m + 10n + 1 = (10m +1)(10n + 1) which if the
number of digits in the number, n + m + 1, are fixed then m and n
are varied over all values with m > n such that their sum is m + n.
The other possible number, N4[m] = 102m + 2*10m + 1, is a perfect square.
Hence there are no palindrome primes with a sum over all digits of 4.

For the sum over all digits of 5 there are two possible types of palindrome numbers.

They are
N5[n,k] = 102n + 10n + 1 + 102nk + 10k ; 1 < k < n–1
and
N5[m] = 102m + 3*10m + 1 .
The number of digits in the first case is 2n + 1 and 2m + 1 in the second
so that multiple primes with the same number of digits are possible
in the first case but only one prime can occur with the same
number of digits. The second case has a slightly larger
percentage of zeros than the first. The results of prime searches
for these two types of palindrome numbers are summarized in the
Tables 2–5 shown below. These primes have the largest percentage
of zeros, the fewest number of primes with the same number of
total digits, the minimum number of different digits (2 and 3), and
except for the primes 11 and 101 the smallest value for the sum
over all integers (5). In the case of N5(n,k) the number of primes
found for a search from n(min) to n(max) which searches
[n(max) – n(min)][n(max) + n(min) – 1]/2 numbers was found to be the
nearly constant value of 136 ± 8 for n(max) – n(min) = 199.
The paucity of multiple palprimes with the same number of digits can
be best appreciated by noting that non-palindrome primes, with 5
ones and the remaining integers all zeros, with only 401 total digits
have 50705 primes with 10467798 numbers searched.


PRIME SEARCHES FOR PRIMES OF THE GENERALIZED FERMAT NUMBERS OF BASE 10

N(a, n) = (a^(2^n)) + 1

with a an integer it can be shown that

N(a, n+k) = (N(a, n) – 1)^(2^k) + 1

  (2^k)-1 
= S[(–1)^j (2^k)! N(a, n)^((2^k) – j)] / [j! ((2^k) – j)!] + 2  
    j = 0 


Note that if a is an odd integer N(a, n) must be even and hence all N(a, n)
are divisible by 2 and no primes are allowed. When a is an even integer N(a, n)
is odd and each N(a, n) has the curious property that each N(a,n) has a unique
set of divisors. In other words, if D is a divisor of N(a,n) then it cannot be
a divisor of N(a,n+k) for all integers k > 0. This does not foreclose on these
numbers being prime. In the case of generalized Fermat numbers of base 10 the
prime search situation is summarized below in Table 1.

Table 1: Smallest divisors of D(k) = (10^(2^k)) + 1
kTest with
PrimeQ[D(k)] (+)
Smallest Divisor of D(k)
A185121
Number of digits in D(k)
A048578 subseq of A000051
1Prime1013
2Prime735
3Not Prime179
4Not Prime35317
5Not Prime1984133
6Not Prime126501107365
7Not Prime257129
8Not Prime10753257
9Not Prime1514497513
10Not Prime18561042846676930571025
11Not Prime1069078036492049
12Not Prime4589240334097
13Not Prime3635898263938497962802538435084289 8193
14Not Prime> 1.37528*(10^11)  (*)16385
15Not Prime6553732769
16Not Prime825753765537
17Not Prime175636481131073
18Not Prime639631361262145
19Not Prime70254593524289
20Not Prime1677721611048577
21(*)> 4.020*(10^9)  (*)2097153
22(*)> 1.000*(10^9)  (*)4194305
23(*)> 5.500*(10^8)  (*)8388609
(+) PrimeQ[N] is the test for primality of the number N in RMathematicaS.
(*) Search limited by computing power.

Conclusion
If primes greater than 101 in D(k) exist then they must have at least 2*(10^6) digits.



Table 2: Statistics for palindrome primes of the type
F(n,k) = (10^2n + 10^n +1) + (10^(2n–k) + 10^k)
Range of
n values
1201-
1400
1401-
1600
1601-
1800
1801-
2000
2001-
2200
2201-
2400
2401-
2600
2601-
2800
#'s
searched
258,700298,500338,300378,100417,900457,700495,000537,300
# of
primes
134140144130127150135131
Density (*)
X 10^4
5.184.694.263.443.043.282.732.44
Range of
# of digits
2603-
2801
2803-
3201
3203-
3601
3603-
4001
4003-
4401
4403-
4801
4803-
5201
5203-
5601
Range of
# of zeros
2398-
2796
2798-
3196
3198-
3596
3598-
3996
3998-
4396
4398-
4796
4798-
5196
5198-
5596
(*)  Density is the ratio of primes to numbers searched for the specified range of n values.



    Table 3: Prime groups with the same number of digits for the
    palindrome primes F(n,k)
    Range of
    n values
    1201-
    1400
    1401-
    1600
    1601-
    1800
    1801-
    2000
    2001-
    2200
    2201-
    2400
    2401-
    2600
    2601-
    2800
    Single prime6772746761696759
    2 primes2018241920182120
    3 primes652761138
    4 primes13412332
    5 primes11000010



      Table 4: First 20 palindrome primes of the
      form G(n) = 10^2n + 3*10^n + 1
      with sum over all digits equal to five.
      n
      A171376
      Number of digits
      A100028
      Number of zeros
      130
      252
      374
      496
      112320
      142926
      163330
      92185182
      133267264
      153307304
      378757754
      448897894
      78515711568
      148829772974
      191538313828
      229745954592
      328665736570
      475595119508
      58251165111648
      78201564115638
      344426888568882
      349416988369880



Table 5: Largest palindrome primes as a function of the number of
primes with the same number of digits for primes of the type F(n,k).
Number of primes
with same # of digits
Number of digitsnkSum over
all digits
15583279112005
2557327861577
2640
5
3560128001050
1476
1752
5
453272663196
1065
1510
2531
5
55195259797
382
534
1187
1852
5



Visualizations

N(n,k) = N( { from 2 to 21 } , 1 )
11111
1101011
110010011
11000100011
1100001000011
110000010000011
11000000100000011
1100000001000000011
 110000000010000000011
11000000000100000000011
1100000000001000000000011
110000000000010000000000011
11000000000000100000000000011
1100000000000001000000000000011
110000000000000010000000000000011
11000000000000000100000000000000011
1100000000000000001000000000000000011
110000000000000000010000000000000000011
11000000000000000000100000000000000000011
1100000000000000000001000000000000000000011

Probable Prime set [ n = 10, 770, 1700, 3411, 3655, … ]

N(n,k) = N( { from 3 to 22 } , 2 )
1011101
101010101
10100100101
1010001000101
 101000010000101
10100000100000101
1010000001000000101
 101000000010000000101
10100000000100000000101
1010000000001000000000101
101000000000010000000000101
10100000000000100000000000101
1010000000000001000000000000101
 101000000000000010000000000000101
10100000000000000100000000000000101
1010000000000000001000000000000000101
101000000000000000010000000000000000101
10100000000000000000100000000000000000101
1010000000000000000001000000000000000000101
 101000000000000000000010000000000000000000101

Probable Prime set [ n = 7, 10, 16, 22, 61, 630, 4315, … ]

N(n,k) = N( { from 4 to 23 } , 3 )
 100111001
10010101001
1001001001001
100100010001001
10010000100001001
1001000001000001001
100100000010000001001
10010000000100000001001
1001000000001000000001001
100100000000010000000001001
10010000000000100000000001001
1001000000000001000000000001001
100100000000000010000000000001001
10010000000000000100000000000001001
1001000000000000001000000000000001001
100100000000000000010000000000000001001
10010000000000000000100000000000000001001
1001000000000000000001000000000000000001001
100100000000000000000010000000000000000001001
10010000000000000000000100000000000000000001001

Probable Prime set
[ n = 4, 73, 79, 168, 303, 370, 384, 484, 1514, 1614, 2833,
5694, 5899, … ]

N(n,k) = N( { from 5 to 24 } , 4 )
10001110001
1000101010001
100010010010001
10001000100010001
1000100001000010001
100010000010000010001
10001000000100000010001
1000100000001000000010001
100010000000010000000010001
10001000000000100000000010001
1000100000000001000000000010001
100010000000000010000000000010001
10001000000000000100000000000010001
1000100000000000001000000000000010001
100010000000000000010000000000000010001
10001000000000000000100000000000000010001
1000100000000000000001000000000000000010001
100010000000000000000010000000000000000010001
10001000000000000000000100000000000000000010001
1000100000000000000000001000000000000000000010001

Probable Prime set [ n = 100, 484, … ]

N(n,k) = N( { from 6 to 25 } , 5 )
1000011100001
100001010100001
10000100100100001
1000010001000100001
100001000010000100001
 10000100000100000100001
1000010000001000000100001
100001000000010000000100001
10000100000000100000000100001
1000010000000001000000000100001
100001000000000010000000000100001
10000100000000000100000000000100001
1000010000000000001000000000000100001
100001000000000000010000000000000100001
10000100000000000000100000000000000100001
1000010000000000000001000000000000000100001
100001000000000000000010000000000000000100001
10000100000000000000000100000000000000000100001
1000010000000000000000001000000000000000000100001
100001000000000000000000010000000000000000000100001

Probable Prime set [ n = 11, 40, 83, 459, 589, 3253, … ]

N(n,k) = N( { from 7 to 25 } , 6 )
100000111000001
 10000010101000001
1000001001001000001
100000100010001000001
10000010000100001000001
1000001000001000001000001
 100000100000010000001000001
10000010000000100000001000001
1000001000000001000000001000001
100000100000000010000000001000001
10000010000000000100000000001000001
 1000001000000000001000000000001000001
100000100000000000010000000000001000001
10000010000000000000100000000000001000001
1000001000000000000001000000000000001000001
100000100000000000000010000000000000001000001
10000010000000000000000100000000000000001000001
1000001000000000000000001000000000000000001000001
100000100000000000000000010000000000000000001000001

Probable Prime set
[ n = 8, 13, 18, 50, 235, 740, 1025, 4373, 4783,
6150, 6680, 8905, … ]

N(n,k) = N( { from 8 to 25 } , 7 )
10000001110000001
1000000101010000001
100000010010010000001
10000001000100010000001
1000000100001000010000001
100000010000010000010000001
10000001000000100000010000001
1000000100000001000000010000001
100000010000000010000000010000001
10000001000000000100000000010000001
1000000100000000001000000000010000001
100000010000000000010000000000010000001
10000001000000000000100000000000010000001
1000000100000000000001000000000000010000001
100000010000000000000010000000000000010000001
10000001000000000000000100000000000000010000001
1000000100000000000000001000000000000000010000001

Probable Prime set [ n = 71, 90, 105, 260, 6942, … ]

N(n,k) = N( { from 9 to 25 } , 8 )
1000000011100000001
100000001010100000001
10000000100100100000001
1000000010001000100000001
100000001000010000100000001
10000000100000100000100000001
1000000010000001000000100000001
100000001000000010000000100000001
10000000100000000100000000100000001
1000000010000000001000000000100000001
100000001000000000010000000000100000001
10000000100000000000100000000000100000001
1000000010000000000001000000000000100000001
100000001000000000000010000000000000100000001
10000000100000000000000100000000000000100000001
1000000010000000000000001000000000000000100000001

Probable Prime set [ n = 103, 295, … ]

N(n,k) = N( { from 10 to 25 } , 9 )
100000000111000000001
10000000010101000000001
1000000001001001000000001
100000000100010001000000001
10000000010000100001000000001
1000000001000001000001000000001
100000000100000010000001000000001
10000000010000000100000001000000001
1000000001000000001000000001000000001
100000000100000000010000000001000000001
10000000010000000000100000000001000000001
1000000001000000000001000000000001000000001
100000000100000000000010000000000001000000001
10000000010000000000000100000000000001000000001
1000000001000000000000001000000000000001000000001

Probable Prime set [ n = 89, 200, 225, 227, 2624, … ]

ps. for k equal from 2 to 9 all searches went up to n = 10000.




N(n) = 102n + 3*10n + 1
N( from 1 to 25 )

 131
 10301
 1003001
 100030001
10000300001
1000003000001
100000030000001
10000000300000001
1000000003000000001
100000000030000000001
 10000000000300000000001
1000000000003000000000001
100000000000030000000000001
 10000000000000300000000000001
1000000000000003000000000000001
 100000000000000030000000000000001
10000000000000000300000000000000001
1000000000000000003000000000000000001
100000000000000000030000000000000000001
10000000000000000000300000000000000000001
1000000000000000000003000000000000000000001
100000000000000000000030000000000000000000001
10000000000000000000000300000000000000000000001
1000000000000000000000003000000000000000000000001
100000000000000000000000030000000000000000000000001






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