Palindromic Primes where the sum over all digits is a minimum

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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^{(2n – k)} + 10^k ; 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] = 10^m + 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] = 10^2m+1 + 1 contains no primes for m > 0.
For even values of m, N₂[2m], can be written as N₂[k,n] = 10^{2^k(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] = 10^{2^k} + 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] = 10^2m + 10^m + 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*10^m+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] = 10^m+n + 10^m + 10ⁿ + 1 = (10^m +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] = 10^2m + 2*10^m + 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 + 10^2n–k + 10^k ; 1 < k < n–1

and

N₅[m] = 10^2m + 3*10^m + 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
k	Test with PrimeQ[D(k)] (+)	Smallest Divisor of D(k) A185121	Number of digits in D(k) A048578 subseq of A000051
1	Prime	101	3
2	Prime	73	5
3	Not Prime	17	9
4	Not Prime	353	17
5	Not Prime	19841	33
6	Not Prime	1265011073	65
7	Not Prime	257	129
8	Not Prime	10753	257
9	Not Prime	1514497	513
10	Not Prime	1856104284667693057	1025
11	Not Prime	106907803649	2049
12	Not Prime	458924033	4097
13	Not Prime	3635898263938497962802538435084289	8193
14	Not Prime	> 1.37528(10^11) ()	16385
15	Not Prime	65537	32769
16	Not Prime	8257537	65537
17	Not Prime	175636481	131073
18	Not Prime	639631361	262145
19	Not Prime	70254593	524289
20	Not Prime	167772161	1048577
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	258,700	298,500	338,300	378,100	417,900	457,700	495,000	537,300
# of primes	134	140	144	130	127	150	135	131
Density ^(*) X 10^4	5.18	4.69	4.26	3.44	3.04	3.28	2.73	2.44
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 prime	67	72	74	67	61	69	67	59
2 primes	20	18	24	19	20	18	21	20
3 primes	6	5	2	7	6	11	3	8
4 primes	1	3	4	1	2	3	3	2
5 primes	1	1	0	0	0	0	1	0

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
1	3	0
2	5	2
3	7	4
4	9	6
11	23	20
14	29	26
16	33	30
92	185	182
133	267	264
153	307	304
378	757	754
448	897	894
785	1571	1568
1488	2977	2974
1915	3831	3828
2297	4595	4592
3286	6573	6570
4755	9511	9508
5825	11651	11648
7820	15641	15638
34442	68885	68882
34941	69883	69880

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 digits	n	k	Sum over all digits
1	5583	2791	1200	5
2	5573	2786	1577 2640	5
3	5601	2800	1050 1476 1752	5
4	5327	2663	196 1065 1510 2531	5
5	5195	2597	97 382 534 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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