Download Analysis of Occurrence of Digit 3 in Prime Numbers till 1 Trillion

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 12, Issue 6 Ver. II (Nov. - Dec.2016), PP 14-21
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Analysis of Occurrence of Digit 3 in Prime Numbers till 1 Trillion
Neeraj Anant Pande
Associate Professor, Department of Mathematics & Statistics,
Yeshwant Mahavidyalaya (College), Nanded – 431602, Maharashtra, INDIA
Abstract: Primes less than 1012 are analyzed for occurrence of digit 3 in them. Multiple occurrences of 3’s are
explored. The first and last occurrences of all multiple 3’s in them are determined within blocks of powers of 10
till 1 trillion.
Keywords: All occurrences, digit 3, Prime numbers.
Mathematics Subject Classification 2010 - 11Y35, 11Y60, 11Y99.
I.
Introduction
It is interesting to see early understanding and exploration of numbers by humans [2]. Many simplest
looking integer types pose great challenges even today; amongst them prime numbers are surely on forefront.
Explorations of primes on theoretical front [1] and in initial vast ranges [4] have just reconfirmed their
irregularities. This paper presents the analysis of occurrence of digit 3 within all primes in ranges of powers of
10 till 1 trillion, i.e., primes p such that 1 < p < 10n, 1 ≤ n ≤ 12. Such analysis for primes is recently done for
digit 0[8], [9], [10], digit 1 [14], [15], [16] and digit 2 [17], [18], [19]. Dividing digits in all natural numbers in
two classes of zero [5], [6], [7] and non-zero [11], [12], [13], such study for them is also available.
II.
Occurrence of Single Digit 3 in Prime Numbers
3 is first odd prime itself. The way in which digit 3 comes in all positive integers can be inferred from work of
[11]. Now all prime numbers p in the range 1 < p < 1012 are considered for trends of occurrences of digit 3.
Table 1: Number of Prime Numbers in Various Ranges with Single 3 in Their Digits
Sr.
No.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
III.
Range
Number of Primes with Single 3
1 – 10
1 – 102
1 – 103
1 – 104
1 – 105
1 – 106
1 – 107
1 – 108
1 – 109
1 – 1010
1 – 1011
1 – 1012
1
9
57
457
3,693
30,928
264,820
2,296,417
20,065,110
176,290,694
1,555,436,420
13,767,790,131
1
Occurrence of Multiple Digit 3’s in Prime Numbers
These results have required long executions on many electronic computers with the aid of efficient
algorithms [3].Single, double, triple and multiple occurrences of digit 3 in all natural numbers in ranges of 1 –
10n is available [11]. The same kind of exploration amongst prime numbers is done here.
Table 2: Number of Prime Numbers in Various Ranges with Multiple 3 in Their Digits
Sr.
No.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Number
Range <
103
104
105
106
107
108
109
1010
1011
1012
DOI: 10.9790/5728-1206021421
Number of Prime
Numbers with 2 3’s
9
92
1,023
10,638
107,948
1,071,488
10,539,313
102,852,807
997,245,608
9,617,083,586
Number of Prime
Numbers with 3 3’s
0
12
154
1,924
23,518
277,526
3,153,036
34,832,837
376,822,954
4,007,995,116
Number of Prime
Numbers with 4 3’s
0
0
7
181
3,023
44,151
596,113
7,624,536
93,623,978
1,113,561,836
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14 | Page
Analysis of Occurrence of Digit 3 in Prime Numbers till 1 Trillion
Table 2: Continued …
Sr.
No.
Number
Range <
106
107
108
109
1010
1011
1012
Number of Prime
Numbers with 5 3’s
14
242
4,440
73,949
1,128,161
16,095,596
217,784,557
Sr.
No.
Number
Range <
Number of
Primes with 8
3’s
8
372
10,056
236,170
1.
2.
3.
4.
5.
6.
7.
Number of Prime
Numbers with 6 3’s
0
13
270
6,171
114,918
1,958,080
30,791,807
Number of Prime
Numbers with 7 3’s
0
0
11
299
7,956
169,209
3,175,699
Table 2: Continued …
1.
2.
3.
4.
109
1010
1011
1012
Number of
Primes with 9
3’s
0
7
388
12,414
Number of
Primes with 10
3’s
0
0
9
411
Number of
Primes with 11
3’s
0
0
0
8
These values of multiple occurrences of digit 3’s in primes in various ranges of 1 - 10i is graphically plotted
with vertical axis in on logarithmic scale.
Figure 1: Number of Primes in Various Ranges with Multiple 3’s in Their Digits
The percentage of primes containing multiple 3’s calculated with respect to number of all such natural numbers
with those many 3’s in respective ranges fluctuates as follows.
Figure2: Percentage of Primes in Various Ranges with Multiple 3’s in Their Digits with Respect to All Such
Integers in Respective Ranges.
DOI: 10.9790/5728-1206021421
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15 | Page
Analysis of Occurrence of Digit 3 in Prime Numbers till 1 Trillion
The peak observed for single 3 in the range 1 – 10 is obvious; the only number 3 with one 3 in this
range is itself a prime, so the percentage becomes 100. There will be no peak thereafter as in any range 1 – 10n
for occurrences of n 3’s, as number having all n digits 3’s will be greater than 3 and divisible by 3 and hence
cannot be prime. We now compare differences of number of multiple occurrences of digits 1 and 2 in prime
numbers with those of 3 in our ranges. Odd man out digit 0 is not considered in these comparisons as it doesn’t
occupy two places in any n digit prime number, viz., units place and leading nth place.
1 Digit Difference with 3
1-10
1 Digit Difference with 3
1-100
0
1 Digit Difference with 3
1-1,000
0
1
0
-1
2
1 Digit Difference with 3
1-10,000
1
0
-5
2
-2
-10
-3
-15
-4
-20
1 Digit Difference with 3
1-100,000
1
0
-20
2
1
-60
-120
-700
-140
-800
-7
-35
-160
-900
5,000
20,000
0
0
-5,000
1
-20,000
2
1 Digit Difference with 3
1-1,000,000,000
1
2
-15,000
-20,000
-60,000
-300,000
-80,000
-400,000
-120,000
-30,000
-140,000
-35,000
-160,000
50,000
-6,000
1 Digit Difference with 3
1-1,000,000,000,000
45,000,000
700,000,000
40,000,000
600,000,000
500,000,000
30,000,000
0
1
2
25,000,000
400,000,000
20,000,000
300,000,000
15,000,000
200,000,000
10,000,000
-100,000
-500,000
-5,000
35,000,000
-50,000
-100,000
-25,000
2
-4,000
1 Digit Difference with 3
1-100,000,000,000
100,000
1
-200,000
-40,000
-10,000
1 Digit Difference with 3
1-10,000,000,000
0
-100,000
-3,000
-600
-30
100,000,000
5,000,000
0
-150,000
-600,000
0
-5,000,000
-700,000
2
-2,000
-500
-25
1
-1,000
-400
-100
-6
1 Digit Difference with 3
1-100,000,000
2
-300
-5
1 Digit Difference with 3
1-10,000,000
1
-200
-80
-2
0
-100
2
-40
-1
1 Digit Difference with 3
1-1,000,000
1
-200,000
-100,000,000
2
1
2
Figure3: Differences of Number of Primes having Single1 and Single2 in their Digits with those having Single3
in them in Ranges of 1 – 10n.
2 Digits Difference with 3
1-100
2
2 Digits Difference with 3
1-1,000
2 Digits Difference with 3
1-10,000
2 Digits Difference with 3
1-100,000
2 Digits Difference with 3
1-1,000,000
3
20
200
1,000
2
10
100
0
0
1
0
-10
-1
1
1
-2
2
-300
-400
-6
-50
-500
-5,000
-7
-60
-600
-6,000
2 Digits Difference with 3
1-100,000,000
5,000
50,000
0
0
-5,000
-15,000
-20,000
-25,000
-30,000
1
2
-50,000
1
-100,000
-150,000
-200,000
-250,000
2
2 Digits Difference with 3
1-1,000,000,000
-4,000
2 Digits Difference with 3
1-100,000,000,000
500,000
5,000,000
50,000,000
0
0
0
-1,000,000
1
2
-5,000,000
-2,000,000
2
-50,000,000
200,000,000
-200,000,000
1
2
-400,000,000
1
2
-600,000,000
-100,000,000
-15,000,000
-20,000,000
2 Digits Difference with 3
1-1,000,000,000,000
0
1
-10,000,000
-1,500,000
2
-3,000
2 Digits Difference with 3
1-10,000,000,000
-500,000
1
-2,000
-40
2 Digits Difference with 3
1-10,000,000
-10,000
2
-30
-5
2
1
-200
-4
1
-100
2
-20
-3
0
1
0
-1,000
-800,000,000
-1,000,000,000
-150,000,000
-35,000
-300,000
-2,500,000
-350,000
-3,000,000
-25,000,000
-200,000,000
-40,000
-45,000
-400,000
-3,500,000
-30,000,000
-250,000,000
-1,200,000,000
-1,400,000,000
-1,600,000,000
-1,800,000,000
Figure4: Differences of Number of Primes having Two1’s and Two2’s in their Digits with those having
Two 3’s in them in Ranges of 1 – 10n.
DOI: 10.9790/5728-1206021421
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16 | Page
Analysis of Occurrence of Digit 3 in Prime Numbers till 1 Trillion
3 Digits Difference with 3
1-1,000
3 Digits Difference with 3
1-10,000
1
3 Digits Difference with 3
1-100,000
0
3 Digits Difference with 3
1-1,000,000
0
1
-2
200
-20
2
1
0
2
-200
-40
-4
-6
1
3 Digits Difference with 3
1-10,000,000
2,000
2
-800
-100
-1,000
-10
-120
-1,200
-12
-140
-1,400
3 Digits Difference with 3
1-100,000,000
3 Digits Difference with 3
1-1,000,000,000
20,000
3 Digits Difference with 3
1-10,000,000,000
200,000
3 Digits Difference with 3
1-100,000,000,000
2,000,000
20,000,000
0
0
0
0
0
-2,000
-20,000
-200,000
-2,000,000
-20,000,000
1
2
1
2
2
-600
-80
-8
0
1
-400
-60
1
2
1
2
-4,000
-40,000
-400,000
-4,000,000
-40,000,000
-6,000
-60,000
-600,000
-6,000,000
-60,000,000
3 Digits Difference with 3
1-1,000,000,000,000
200,000,000
0
1
2
-200,000,000
1
2
-400,000,000
-600,000,000
-8,000
-80,000
-800,000
-8,000,000
-80,000,000
-10,000
-100,000
-1,000,000
-10,000,000
-100,000,000
-12,000
-120,000
-1,200,000
-12,000,000
-120,000,000
-1,000,000,000
-14,000
-140,000
-1,400,000
-14,000,000
-140,000,000
-1,200,000,000
-16,000
-160,000
-1,600,000
-16,000,000
-160,000,000
-1,400,000,000
-800,000,000
Figure5: Differences of Number of Primes having Three1’s and Three2’s in their Digits with those having
Three3’s in them in Ranges of 1 – 10n.
4 Digits Difference with 3
1-10,000
4 Digits Difference with 3
1-100,000
1
4 Digits Difference with 3
1-1,000,000
4
20
3
0
2
-20
1
0
1
-2
2
-80
-3
-100
-4
4 Digits Difference with 3
1-10,000,000
500
0
-500
-1,000
-1,500
-2,000
-2,500
2
-6
-140
-7
-160
4 Digits Difference with 3
1-100,000,000
4 Digits Difference with 3
1-1,000,000,000
5,000
50,000
0
0
0
-50,000
-500,000
-5,000
1
-120
-5
1
2
1
-10,000
-15,000
-20,000
2
4 Digits Difference with 3
1-10,000,000,000
500,000
1
2
-60
-1
0
1
-40
2
-100,000
4 Digits Difference with 3
1-100,000,000,000
4 Digits Difference with 3
1-1,000,000,000,000
10,000,000
100,000,000
1
-1,000,000
2
-10,000,000
-1,500,000
-150,000
-3,500,000
-30,000
-350,000
-4,000,000
-35,000
-400,000
-4,500,000
-100,000,000
1
2
-300,000,000
-30,000,000
-3,000,000
-300,000
2
-20,000,000
-2,500,000
-250,000
-25,000
1
-200,000,000
-2,000,000
-200,000
0
0
-400,000,000
-40,000,000
-500,000,000
-50,000,000
-600,000,000
Figure6: Differences of Number of Primes having Four1’s and Four2’s in their Digits with those having
Four 3’s in them in Ranges of 1 – 10n.
5 Digits Difference with 3
1-100,000
1
5 Digits Difference with 3
1-1,000,000
0
-2
1
2
-4
-6
-8
-10
-12
0
-14
1
DOI: 10.9790/5728-1206021421
2
-16
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Analysis of Occurrence of Digit 3 in Prime Numbers till 1 Trillion
5 Digits Difference with 3
1-10,000,000
5 Digits Difference with 3
1-100,000,000
0
5 Digits Difference with 3
1-1,000,000,000
500
1
-50
10,000
0
2
1
2
1
2
2,000,000
20,000,000
1
2
-2,000,000
-1,500
-20,000
-300,000
-4,000,000
-2,000
-30,000
-400,000
-6,000,000
1
-20,000,000
2
-50,000
-4,000
-700,000
-60,000
2
-80,000,000
-8,000,000
-100,000,000
-600,000
-3,500
1
-40,000,000
-60,000,000
-500,000
-40,000
0
0
-200,000
-3,000
-250
100,000
0
-10,000
-2,500
-200
5 Digits Difference with 3
1-1,000,000,000,000
-100,000
-1,000
-150
5 Digits Difference with 3
1-100,000,000,000
0
-500
-100
5 Digits Difference with 3
1-10,000,000,000
-800,000
-10,000,000
-120,000,000
-12,000,000
-140,000,000
Figure7: Differences of Number of Primes having Five1’s and Five2’s in their Digits with those having Five 3’s
in them in Ranges of 1 – 10n.
6 Digits Difference with 3
1-1,000,000
1
0
1
6 Digits Difference with 3
1-10,000,000
6 Digits Difference with 3
1-100,000,000
0
6 Digits Difference with 3
1-1,000,000,000
0
-2
1
2
-50
2
6 Digits Difference with 3
1-10,000,000,000
0
1
2
10,000
1
-1,000
-2,000
-6
1
-8
2
-200
-14
-250
-5,000,000
-1,000,000
-80,000
-90,000
-1,400,000
-6,000
-100,000
-1,600,000
2
-15,000,000
-1,200,000
-5,000
1
-10,000,000
-800,000
-60,000
-70,000
-12
2
-600,000
-50,000
-4,000
1
-400,000
-40,000
-150
-10
0
-200,000
-30,000
-3,000
5,000,000
0
-10,000
-100
6 Digits Difference with 3
1-1,000,000,000,000
200,000
0
2
-20,000
-4
6 Digits Difference with 3
1-100,000,000,000
-20,000,000
-25,000,000
Figure8: Differences of Number of Primes having Six1’s and Six2’s in their Digits with those having Six 3’s in
them in Ranges of 1 – 10n.
7 Digits Difference with 3
1-10,000,000
1
7 Digits Difference with 3
1-100,000,000
7 Digits Difference with 3
1-1,000,000,000
7 Digits Difference with 3
1-10,000,000,000
7 Digits Difference with 3
1-100,000,000,000
6
50
1,000
20,000
0
0
0
4
2
-50
0
-2
1
1
-1,000
2
-100
2
-4
-150
-6
-10
1
2
-12
-20,000
2
-40,000
-3,000
-60,000
-4,000
-80,000
-5,000
-100,000
-250
-6,000
-120,000
-300
-7,000
-140,000
-200
-8
0
1
-2,000
7 Digits Difference with 3
1-1,000,000,000,000
500,000
0
1
2
-500,000
1
2
-1,000,000
-1,500,000
-2,000,000
-2,500,000
Figure9: Differences of Number of Primes having Seven1’s and Seven2’s in their Digits with those having
Seven 3’s in them in Ranges of 1 – 10n.
8 Digits Difference with 3
1-100,000,000
1
8 Digits Difference with 3
1-1,000,000,000
8 Digits Difference with 3
1-10,000,000,000
3
50
2
-50
-100
-3,000
-150
-4,000
-4
-200
-5,000
-5
-250
-6
2
50,000
0
-3
1
1,000
-1,000
-1
-2
0
8 Digits Difference with 3
1-1,000,000,000,000
0
0
1
8 Digits Difference with 3
1-100,000,000,000
-7
-8
1
2
1
2
-2,000
0
1
2
-6,000
-7,000
-300
-8,000
-350
-9,000
-50,000
1
2
-100,000
-150,000
-200,000
-250,000
Figure10: Differences of Number of Primes having Eight1’s and Eight2’s in their Digits with those having
Eight 3’s in them in Ranges of 1 – 10n.
DOI: 10.9790/5728-1206021421
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18 | Page
Analysis of Occurrence of Digit 3 in Prime Numbers till 1 Trillion
9 Digits Difference with 3
1-1,000,000,000
9 Digits Difference with 3
1-10,000,000,000
1
9 Digits Difference with 3
1-100,000,000,000
6
0
4
-50
9 Digits Difference with 3
1-1,000,000,000,000
2,000
1
0
2
-100
2
-2,000
-150
0
1
2
-4,000
-200
-2
1
2
0
1
2
-6,000
-250
-4
-8,000
-300
-6
-350
-10,000
-8
-400
-12,000
Figure11: Differences of Number of Primes having Nine1’s and Nine2’s in their Digits with those having
Nine 3’s in them in Ranges of 1 – 10n.
10 Digits Difference with 3
1-10,000,000,000
10 Digits Difference with 3
1-100,000,000,000
1
10 Digits Difference with 3
1-1,000,000,000,000
4
50
0
2
-50
0
-2
-100
1
1
2
-150
2
-200
0
-4
-250
-6
-300
-350
-8
1
2
-400
-10
-450
Figure12: Differences of Number of Primes having Ten1’s and Ten2’s in their Digits with those having Ten 3’s
in them in Ranges of 1 – 10n.
11 Digits Difference with 3
1-100,000,000,000
11 Digits Difference with 3
1-1,000,000,000,000
1
6
4
2
0
-2
1
2
-4
-6
0
-8
1
2
-10
Figure13: Differences of Number of Primes having Eleven1’s and Eleven2’s in their Digits with those having
Eleven 3’s in them in Ranges of 1 – 10n.
12 Digits Difference with 3
1-1,000,000,000,000
1
0
1
2
Figure14: Differences of Number of Primes having Twelve1’s and Twelve2’s in their Digits with those having
Twelve 3’s in them in Ranges of 1 – 10n.
IV.
First Occurrence of Digit 3 in Prime Numbers
The first positive integer with single digit 3 is 3 itself! For sufficiently large ranges, first positive
integer containing 2 3’s 33, 3 3’s is 333 and so on. Simple formulation for this is generalized in [11].
Formula 1 [11] : If n and r are natural numbers, then the first occurrence of r number of 3’s in numbers in range
1 ≤ m < 10n is

, if r  n

 r 1
f= 
.
j
 3 10 , if r  n
 j 0


DOI: 10.9790/5728-1206021421
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19 | Page
Analysis of Occurrence of Digit 3 in Prime Numbers till 1 Trillion
This was for all natural numbers. The first occurrences of r number of 3’s in prime numbers in range
1 ≤ m < 10n doesn’t have a common formula yet. Owing to this they have actually been determined.
Table 3: First Prime Numbers in Various Ranges with Multiple 3’s in Their Digits
Sr.
No.
Range
13
3
3
3
3
3
3
3
3
3
3
3
3
1 – 101
1 – 102
1 – 103
1 – 104
1 – 105
1 – 106
1 – 107
1 – 108
1 – 109
1 – 1010
1 – 1011
1 – 1012
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
2 3’s
233
233
233
233
233
233
233
233
233
233
First Prime Number in Range with
3 3’s
4 3’s
5 3’s
6 3’s
2,333
2,333
23,333
2,333
23,333
313,333
2,333
23,333
313,333
3,233,333
2,333
23,333
313,333
3,233,333
2,333
23,333
313,333
3,233,333
2,333
23,333
313,333
3,233,333
2,333
23,333
313,333
3,233,333
2,333
23,333
313,333
3,233,333
7 3’s
31,333,333
31,333,333
31,333,333
31,333,333
31,333,333
Table 3: Continued
Sr.
No.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
First Prime Number in Range with
8 3’s
9 3’s
333,233,333
333,233,333
3,233,333,333
333,233,333
3,233,333,333
333,233,333
3,233,333,333
Range
1 – 101
1 – 102
1 – 103
1 – 104
1 – 105
1 – 106
1 – 107
1 – 108
1 – 109
1 – 1010
1 – 1011
1 – 1012
V.
10 3’s
23,333,333,333
23,333,333,333
11 3’s
333,313,333,333
Last Occurrence of Digit 3 in Prime Numbers
The largest natural number with r number of 3’s in its digits in ranges 1 – 10n, 1 ≤ n ≤ 12, fits in a formula.
Formula 2 [11] : If n and r are natural numbers, then the last occurrence of r number of 3’s in numbers in range
1 ≤ m < 10n is

, if r  n


0
, if r  n


l =  r 1
.
 n 1
j
j
 3 10  
 9  10 , if r  n
 j 0
 j r

Since such primes don’t fit in any formula, the last prime numbers with r number of 3’s in them in ranges 1 –
10n, 1 ≤ n ≤ 12, have been computationally determined.




Table 4: Last Prime Numbers in Various Ranges with Multiple 3’s in Their Digits
Sr.
No.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Number
of 3’s
1
2
3
4
5
6
7
8
9
10
11
Last Prime Number in Range 1 –
101
102
103
104
3
83
983
9,973
733
9,833
7,333
-
DOI: 10.9790/5728-1206021421
105
99,923
99,833
93,383
38,333
-
106
999,983
999,433
997,333
973,333
733,333
-
107
9,999,973
9,999,533
9,998,333
9,943,333
9,533,333
3,733,333
-
www.iosrjournals.org
108
99,999,931
99,999,373
99,993,833
99,983,333
99,338,333
93,733,333
83,333,333
-
20 | Page
Analysis of Occurrence of Digit 3 in Prime Numbers till 1 Trillion
Table 4: Continued …
Sr.
No.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Number
of 3’s
1
2
3
4
5
6
7
8
9
10
11
Last Prime Number in Range 1 –
109
1010
999,999,937
9,999,999,943
999,999,733
9,999,999,833
999,993,833
9,999,993,833
999,935,333
9,999,937,333
999,343,333
9,999,533,333
995,333,333
9,995,333,333
983,333,333
9,943,333,333
373,333,333
9,433,333,333
3,334,333,333
-
1011
99,999,999,943
99,999,999,833
99,999,993,833
99,999,933,433
99,999,335,333
99,994,333,333
99,934,333,333
99,433,333,333
97,333,333,333
38,333,333,333
-
Remark : The maximum number of 3’s in any prime in the range 1 – 10n is at most n − 1, except n = 1.
The numbers coming in all sections of this work give new integer sequences and which are important enough to
merit independent analysis.
Acknowledgements
This work required enormous computational work which could be made possible by using Java
Programming Language, NetBeans IDE, Microsoft Office Excel on software side, for which their Development
Teams deserve gratitude, and on hardware side, all the computers of Laboratory of Mathematics & Statistics
Department of the author’s institution were employed for very long durations and are duly acknowledged.
Additional support of the uninterrupted power supply given by the Department of Electronics of the institute
made these runs smooth. The author expresses gratitude to the University Grants Commission (U.G.C.), New
Delhi of the Government of India for funding a Research Project (F.No. 47-748/13(WRO)) to the author, which
was instrumental in setting out initial ideas of this work. The author is also thankful to the anonymous referees
of this paper.
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[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
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DOI: 10.9790/5728-1206021421
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