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Address： School of Information Science and Engineering， Southeast University， Nanjing， PR. China； E-mail： [email protected]
Received： November 14， 2013/ Accepted： January 5， 2014/ Published online：
February 26， 2014
Abstract： This paper presents an approximate expression of the number of Mersenne primes on the basis of Zhou’s conjecture， predicts the position of each Mersenne prime， and compares the number of Mersenne primes derived from this expression to the number of Mersenne primes derived from actual values and the four existing approximate expressions.
Key words： Mersenne primes； Zhou’s conjecture； Distribution； Approximate expression
Du， W. L.， Shen， B.， & Zhang， Y. Q. （2014）. Research on the Distribution of Mersenne Primes Based on Zhou’s Conjecture. Studies in Mathematical Sciences， 8（1），-0. Available from URL： http：//www.cscanada.net/index.php/sms/article/view/3976 DOI： http：//dx.doi.org/10.3968/3976
1. INTRODUCTION
In the modern number theory， research on Mersenne primes is an important topic， although it has been investigated for over two thousand years. The grid technology has greatly improved the research on Mersenne primes， and has resulted in the discovery of the forty-eighth Mersenne prime in 2013.
One of the important aspects to research the Mersenne prime is the study of its distribution. Many mathematicians have given their proposed conjectures on it. Regretfully， those conjectures are well have obvious deviation from the actual distribution. This paper presents an approximate expression of Mersenne primes based on the precise expression known as Zhou’s conjecture， named after Chinese mathematician Haizhong Zhou ， who first proposed this conjecture in 1992.
2. MERSENNE PRIMES AND ZHOU’S CONJECTURE
A prime number is a natural number greater than 1， which does not have any positive divisors other than 1 and itself[1]. There are infinite prime numbers. A Mersenne prime is a prime number of the form MP=2p-1， where p is an assumed prime. They are called after the French mathematician Marin Mersenne who studied them in the early 17th century. Research on Mersenne primes is an essential topic of modern number theory. The study of Mersenne prime has been showed hot but difficult.
The Lucas–Lehmer test is a primality test for Mersenne numbers. The test was originally developed by French mathematician Fran?ois Lucas in 1856[2]， subsequently improved by Lucas in 1878 and American mathematician Derrick Lehmer in the 1930s. Lucas-Lehmer test is that ， for p an odd prime， the Mersenne number 2p-1 is prime if and only if 2p-1 divides S（p ? 1） where S（p ? 1） defined by induction as S（n + 1） = （S （n））2 ? 2， and S（1） = 4. This approach is still playing an important role in computation method. In 1952， American mathematician Raphael Robinson compiled computer program based on Lucas–Lehmer test. As of February 2013， 48 Mersenne primes are known. M25，964，951 has been found to be the 42th Mersenne prime. In modern times， the largest known prime has almost always been a Mersenne prime.
Mersenne primes distribution is extremely irregular， which leads to the irregular time intervals to discover Mersenne primes[3]. Exploring the distribution of Mersenne primes seems to be more difficult than finding new Mersenne primes. Many fundamental questions about Mersenne primes remain unresolved. It is even unknown whether the number of Mersenne primes is finite or infinite， on which for a long time Mathematicians have made some conjectures.
Alexander Hurwitz reported that he had applied Lucas’ test to investigate the primality of the Mersenne Numbers between 3，300 < p < 5，000 and discovered that M4253 and M4423 are prime numbers[4]. American mathematician Daniel Shanks thought that the approximate number of Mersenne primes is in the range pn ≤ p ≤ pm. So he thought there are about five Mersenne primes in the interval 500≤ p ≤50，000[3]. Actually， seven Mersenne primes was noted in that interval.
Canadian mathematician Donald Gillies guessed that there are about two Mersenne primes when p is between x and 2x. x is a positive integer greater than one[5]. But the distribution of Mersenne primes derived from Donald Gillies’ conjecture is quite different from the actual distribution. Sometimes， none of Mersenne prime can be found between x and 3x， or even between x and 4x. For example， none of Mersenne prime exists between x and 3x when x = 22000. There is not any Mersenne prime between x and 4x when x = 128.
American mathematician John Brillhart[3]also made a conjecture which stated that p of the n-th Mersenne primes Mp is approximately equal to （1.5）n. But the distribution of Mersenne primes derived from John Brillhart’s conjecture is still quite different from the actual distribution.
Dutch mathematician Hendrik Lenstra and American mathematician Carl Pomerance gave a conjecture on the number of Mersenne primes in 1980 which can be repressed as the following three statements. The restated form is indicated by American mathematician Samuel S. Wagstaff in 1983[6].
The number of Mersenne primes less than or equal to x is about ， where γ is Euler’s constant.
The expected number of Mersenne primes 2p-1 with p between x and 2x is about eγ. The probability that 2p-1 being a prime is about where a=2 if p=3 （mod 4）， and a=6 if p=1 （mod 4）.
In 1992， Haizhong Zhou gave a precise expression about Mersenne primes’ distribution[7]. This important achievement has been named as “Zhou’s conjecture”. In fact， Zhou’s conjecture is important and helpful to research the distribution of Mersenne primes.
Zhou’s conjecture states that if 22n 3. THE MAIN RESEARCH RESULTS
Based on Zhou’s conjecture， we can deduce that the number of Mersenne primes is ， where . Until now， there are 5 data which is verified to be the actual numbers： . Our conjecture numbers are coincident with the actual numbers in these five data.
When N is an arbitrary natural number except 1， Table 1 gives the comparison of definite numbers with the numbers of our conjecture and Lenstra and Pomerance’s conjecture.
Table 1
Mersenne Prime Number Forecast
Actual numbers The exponent p Our conjecture numbers Our conjecture’s deviation L&P’s conjecture numbers L&P’s conjecture deviation
1 2 1 0 0.8 0.2
2 3 1.5 0.5 1.9 0.1
3 5 2.4 0.6 3.2 -0.2
4 7 3.1 0.9 4.1 -0.1
5 13 4.5 0.5 5.6 -0.6
6 17 5.2 0.8 6.3 -0.3
7 19 5.4 1.6 6.6 0.4
8 31 6.6 1.4 7.9 0.1
9 61 8.3 0.7 9.6 -0.6
10 89 9.3 0.7 10.6 -0.6
11 107 9.8 1.2 11.1 -0.1
12 127 10.2 1.8 11.5 0.5
13 521 13.9 -0.9 15.1 -2.1
14 607 14.3 -0.3 15.5 -1.5
15 1279 16.3 -1.3 17.4 -2.4
16 2203 17.7 -1.7 18.8 -2.8
17 2281 17.8 -0.8 18.9 -1.9
18 3217 18.8 -0.8 19.8 -1.8
19 4253 19.5 -0.5 20.5 -1.5
20 4423 19.6 0.4 20.6 -0.6
21 9689 21.8 -0.8 22.6 -1.6
22 9941 21.8 0.2 22.7 -0.7
23 11213 22.2 0.8 23.0 0.0
24 19937 23.7 0.3 24.5 -0.5
25 21701 24.0 1.0 24.7 0.3
26 23209 24.2 1.8 24.9 1.1
27 44497 25.9 1.1 26.6 0.4
28 86243 27.8 0.2 28.3 -0.3
29 110503 28.4 0.6 28.9 0.1
30 132049 28.9 1.1 29.4 0.6
31 216091 30.3 0.7 30.6 0.4
32 756839 33.8 -1.8 33.8 -1.8
33 859433 34.1 -1.1 34.2 -1.2
34 1257787 35.2 -1.2 35.2 -1.2
35 1398269 35.5 -0.5 35.4 -0.4
36 2976221 37.6 -1.6 37.4 -1.4 37 3021377 37.6 -0.6 37.4 -0.4
38 6972593 40.0 -2.0 39.6 -1.6
39 13466917 41.8 -2.8 41.2 -2.2
40 20996011 43.1 -3.1 42.4 -2.4
41 24036583 43.4 -2.4 42.7 -1.7
42 25964951 43.6 -1.6 42.9 -0.9
43* 30402457 44.1 -1.1 43.3 -0.3
44* 32582657 44.3 -0.3 43.5 0.5
45* 37156667 44.6 0.4 43.9 1.1
46* 42643801 45.0 1.0 44.2 1.8
47* 43112609 45.1 1.9 44.2 2.8
48* 57885161 45.9 2.1 45.0 3.0
Note. Our conjecture numbers：
Our conjecture’s deviation：
L&P’s conjecture numbers： L&P’s conjecture deviation：
* It is uncertain that whether any undiscovered Mersenne primes exist between the 42nd （M25，964，951） and the 48th （M57，885，161） in this Table. Therefore the order is provisional.
Theory number of our conjecture in this paper is very similar to the actual number. There are 25 numbers with deviation less than 1， but only one number of deviation greater than 3 where p is equal to 20，996，011. The first 42 numbers are determined but the last 6 numbers are undetermined. For the first 38 Mersenne primes， the maximum deviation of our conjecture is smaller than Lenstra and Pomerance’s conjecture. If there is one or more other Mersenne primes when 25，964，951 In the interval of [N1， N2]， the number of Mersenne primes is where . N1 and N2 can be extended to the whole natural number， but the precision of this formula will quickly decrease with the reduction of .
The number of Mersenne primes is
， when 5，000≤p≤50，000. The actual number is seven . The number in this paper comes closer to the actual number than Shanks’ conjecture.
Since the approximate number of Mersenne prime is
n=21og2N-log2log2N-1， then 2log2N=n+1+log2log2N， hence . According to， p of the nth Mersenne prime is . Actual values of p are compared with the values in our conjecture and Brillhart’s conjecture showed in Table 2.
Table 2
Values of p for n-th Mersenne Primes
Number The exponent p Our conjecture Our conjecture’s ratio Brillhart’s conjecture value Brillhart’s conjecture Ratio
1 2 2 1.0 1.5 1.3
2 3 3 1.0 2 1.5 3 5 6 0.8 3 1.7
4 7 9 0.8 5 1.4
5 13 14 0.9 8 1.6
6 17 21 0.8 11 1.5
7 19 32 0.6 17 1.1
8 31 48 0.6 26 1.2
9 61 72 0.8 38 1.6
10 89 106 0.8 58 1.5
11 107 157 0.7 86 1.2
12 127 231 0.5 130 1.0
13 521 339 1.5 195 2.7
14 607 496 1.2 292 2.1
15 1279 724 1.8 438 2.9
16 2203 1056 2.1 657 3.4
17 2281 1536 1.5 985 2.3
18 3217 2232 1.4 1478 2.2
19 4253 3238 1.3 2217 1.9
20 4423 4693 0.9 3325 1.3
21 9689 6792 1.4 4988 1.9
22 9941 9822 1.0 7482 1.3
23 11213 14189 0.8 11223 1.0
24 19937 20480 1.0 16834 1.2
25 21701 29537 0.7 25251 0.9
26 23209 42566 0.5 37877 0.6
27 44497 61303 0.7 56815 0.8
28 86243 88231 1.0 85223 1.0
29 110503 126910 0.9 127834 0.9
30 132049 182445 0.7 191751 0.7
31 216091 262144 0.8 287627 0.8
32 756839 376476 2.0 431440 1.8
33 859433 540424 1.6 647160 1.3
34 1257787 775432 1.6 970740 1.3
35 1398269 1112183 1.3 1456110 1.0
36 2976221 1594560 1.9 2184164 1.4
37 3021377 2285318 1.3 3276247 0.9
38 6972593 3274178 2.1 4914370 1.4
39 13466917 4689375 2.9 7371555 1.8
40 20996011 6714162 3.1 11057332 1.9
41 24036583 9610358 2.5 16585998 1.4
42 25964951 13751945 1.9 24878998 1.0
43* 30402457 19673030 1.5 37318497 0.8
44* 32582657 28136247 1.2 55977745 0.6
45* 37156667 40230350 0.9 83966617 0.4
46* 42643801 57509400 0.7 125949930 0.3
47* 43112609 82191237 0.5 188924890 0.2
48* 57885161 117440510 0.5 283387330 0.2
Note. Our conjecture： Our conjecture ratio ：
Brillhart’s conjecture value ： （1.5）n Brillhart’s conjecture ratio ：
Values in this paper and Brillhart’s conjecture have a certain gap with the actual values. Conjecture values are not more than the twice of actual values most of the time. Actual values are most 2.1 times of the conjecture values in this paper when p ≤ 6，972，593. Actual values are right on most 3.4 times of the Brillhart’s conjecture values. The first 42 ratios have been determined and the last 6 ratio are undetermined. If there is no other Mersenne prime when p < 57，885，161， Brillhart’s conjecture values are at most 5 times than actual values for the last 10 Mersenne primes. Nonetheless， actual values are at most 3.1 times than conjecture values in this paper. If there is one or more other Mersenne primes when p < 57，885，161， Brillhart’s conjecture values are at most 8 times than actual values for the last 10 Mersenne primes. But actual values are still at most 3.1 times than conjecture values in this paper. From the above deducing， we can gauss that there are 2 Mersenne primes in average when x ≤ p ≤ 2x where x is large. Actually， sometimes， there is no Mersenne prime between x and 2x. But sometimes there are more than one Mersenne prime. For example， seven Mersenne primes exist at least when x is 24，036，583. When x is smaller， the average of Mersenne primes number is less than 2 between x and 2x.
The probability that MN is a Mersenne prime is
.
So the probability that Mp is a Mersenne prime is
.
4. SUMMARY
It is difficult to find the pattern of Mersenne primes because the distribution of Mersenne primes is very irregular. Based on Zhou’s conjecture， this paper uses formula-transformation method， and derives the approximated expression of Mersenne primes. The numbers of Mersenne primes estimated by this expression are much nearer to the actual numbers than the numbers estimated by Lenstra and Pomerance. The position of each Mersenne prime is also of the view by this expression and the results are better than the estimation in Brillharts’ conjecture. The results in this paper are closer to the actual value than Shanks and Gillies’ conjecture.
REFERENCES
Zhang， S. B.， & Chen， X. M. （2013）. Mersenne primes and Zhou’s conjecture. Science & Technology Review， 31（3）， 84.
Lehmer， D. H. （1935）. On Lucas’test for primality of Mersenne’s numbers. J. of the London Math. Soc.， 10， 162-165.
Zhang S. B. （2008）. Review of research on Mersenne primes. Science & Technology Review， 26（18）， 88-92.
Hurwitz， A. （1962）. New Mersenne primes. Math.Comp， 16， 249-251.
Gillies， D. B. （1964）. Three new Mersenne primes and a statistical theory. Math Comp， 18， 93-97.
Wagstaff， S. S. （1983）. Divisors of Mersenne numbers. Math Comp， 40， 385-397.
Zhou， H. Z. （1992）. The distribution of Mersenne primes. Acta Scientiarum Naturalicm Universitatis Sunyatseni， 31（4）， 121-122.
Received： November 14， 2013/ Accepted： January 5， 2014/ Published online：
February 26， 2014
Abstract： This paper presents an approximate expression of the number of Mersenne primes on the basis of Zhou’s conjecture， predicts the position of each Mersenne prime， and compares the number of Mersenne primes derived from this expression to the number of Mersenne primes derived from actual values and the four existing approximate expressions.
Key words： Mersenne primes； Zhou’s conjecture； Distribution； Approximate expression
Du， W. L.， Shen， B.， & Zhang， Y. Q. （2014）. Research on the Distribution of Mersenne Primes Based on Zhou’s Conjecture. Studies in Mathematical Sciences， 8（1），
1. INTRODUCTION
In the modern number theory， research on Mersenne primes is an important topic， although it has been investigated for over two thousand years. The grid technology has greatly improved the research on Mersenne primes， and has resulted in the discovery of the forty-eighth Mersenne prime in 2013.
One of the important aspects to research the Mersenne prime is the study of its distribution. Many mathematicians have given their proposed conjectures on it. Regretfully， those conjectures are well have obvious deviation from the actual distribution. This paper presents an approximate expression of Mersenne primes based on the precise expression known as Zhou’s conjecture， named after Chinese mathematician Haizhong Zhou ， who first proposed this conjecture in 1992.
2. MERSENNE PRIMES AND ZHOU’S CONJECTURE
A prime number is a natural number greater than 1， which does not have any positive divisors other than 1 and itself[1]. There are infinite prime numbers. A Mersenne prime is a prime number of the form MP=2p-1， where p is an assumed prime. They are called after the French mathematician Marin Mersenne who studied them in the early 17th century. Research on Mersenne primes is an essential topic of modern number theory. The study of Mersenne prime has been showed hot but difficult.
The Lucas–Lehmer test is a primality test for Mersenne numbers. The test was originally developed by French mathematician Fran?ois Lucas in 1856[2]， subsequently improved by Lucas in 1878 and American mathematician Derrick Lehmer in the 1930s. Lucas-Lehmer test is that ， for p an odd prime， the Mersenne number 2p-1 is prime if and only if 2p-1 divides S（p ? 1） where S（p ? 1） defined by induction as S（n + 1） = （S （n））2 ? 2， and S（1） = 4. This approach is still playing an important role in computation method. In 1952， American mathematician Raphael Robinson compiled computer program based on Lucas–Lehmer test. As of February 2013， 48 Mersenne primes are known. M25，964，951 has been found to be the 42th Mersenne prime. In modern times， the largest known prime has almost always been a Mersenne prime.
Mersenne primes distribution is extremely irregular， which leads to the irregular time intervals to discover Mersenne primes[3]. Exploring the distribution of Mersenne primes seems to be more difficult than finding new Mersenne primes. Many fundamental questions about Mersenne primes remain unresolved. It is even unknown whether the number of Mersenne primes is finite or infinite， on which for a long time Mathematicians have made some conjectures.
Alexander Hurwitz reported that he had applied Lucas’ test to investigate the primality of the Mersenne Numbers between 3，300 < p < 5，000 and discovered that M4253 and M4423 are prime numbers[4]. American mathematician Daniel Shanks thought that the approximate number of Mersenne primes is in the range pn ≤ p ≤ pm. So he thought there are about five Mersenne primes in the interval 500≤ p ≤50，000[3]. Actually， seven Mersenne primes was noted in that interval.
Canadian mathematician Donald Gillies guessed that there are about two Mersenne primes when p is between x and 2x. x is a positive integer greater than one[5]. But the distribution of Mersenne primes derived from Donald Gillies’ conjecture is quite different from the actual distribution. Sometimes， none of Mersenne prime can be found between x and 3x， or even between x and 4x. For example， none of Mersenne prime exists between x and 3x when x = 22000. There is not any Mersenne prime between x and 4x when x = 128.
American mathematician John Brillhart[3]also made a conjecture which stated that p of the n-th Mersenne primes Mp is approximately equal to （1.5）n. But the distribution of Mersenne primes derived from John Brillhart’s conjecture is still quite different from the actual distribution.
Dutch mathematician Hendrik Lenstra and American mathematician Carl Pomerance gave a conjecture on the number of Mersenne primes in 1980 which can be repressed as the following three statements. The restated form is indicated by American mathematician Samuel S. Wagstaff in 1983[6].
The number of Mersenne primes less than or equal to x is about ， where γ is Euler’s constant.
The expected number of Mersenne primes 2p-1 with p between x and 2x is about eγ. The probability that 2p-1 being a prime is about where a=2 if p=3 （mod 4）， and a=6 if p=1 （mod 4）.
In 1992， Haizhong Zhou gave a precise expression about Mersenne primes’ distribution[7]. This important achievement has been named as “Zhou’s conjecture”. In fact， Zhou’s conjecture is important and helpful to research the distribution of Mersenne primes.
Zhou’s conjecture states that if 22n
Based on Zhou’s conjecture， we can deduce that the number of Mersenne primes is ， where . Until now， there are 5 data which is verified to be the actual numbers： . Our conjecture numbers are coincident with the actual numbers in these five data.
When N is an arbitrary natural number except 1， Table 1 gives the comparison of definite numbers with the numbers of our conjecture and Lenstra and Pomerance’s conjecture.
Table 1
Mersenne Prime Number Forecast
Actual numbers The exponent p Our conjecture numbers Our conjecture’s deviation L&P’s conjecture numbers L&P’s conjecture deviation
1 2 1 0 0.8 0.2
2 3 1.5 0.5 1.9 0.1
3 5 2.4 0.6 3.2 -0.2
4 7 3.1 0.9 4.1 -0.1
5 13 4.5 0.5 5.6 -0.6
6 17 5.2 0.8 6.3 -0.3
7 19 5.4 1.6 6.6 0.4
8 31 6.6 1.4 7.9 0.1
9 61 8.3 0.7 9.6 -0.6
10 89 9.3 0.7 10.6 -0.6
11 107 9.8 1.2 11.1 -0.1
12 127 10.2 1.8 11.5 0.5
13 521 13.9 -0.9 15.1 -2.1
14 607 14.3 -0.3 15.5 -1.5
15 1279 16.3 -1.3 17.4 -2.4
16 2203 17.7 -1.7 18.8 -2.8
17 2281 17.8 -0.8 18.9 -1.9
18 3217 18.8 -0.8 19.8 -1.8
19 4253 19.5 -0.5 20.5 -1.5
20 4423 19.6 0.4 20.6 -0.6
21 9689 21.8 -0.8 22.6 -1.6
22 9941 21.8 0.2 22.7 -0.7
23 11213 22.2 0.8 23.0 0.0
24 19937 23.7 0.3 24.5 -0.5
25 21701 24.0 1.0 24.7 0.3
26 23209 24.2 1.8 24.9 1.1
27 44497 25.9 1.1 26.6 0.4
28 86243 27.8 0.2 28.3 -0.3
29 110503 28.4 0.6 28.9 0.1
30 132049 28.9 1.1 29.4 0.6
31 216091 30.3 0.7 30.6 0.4
32 756839 33.8 -1.8 33.8 -1.8
33 859433 34.1 -1.1 34.2 -1.2
34 1257787 35.2 -1.2 35.2 -1.2
35 1398269 35.5 -0.5 35.4 -0.4
36 2976221 37.6 -1.6 37.4 -1.4 37 3021377 37.6 -0.6 37.4 -0.4
38 6972593 40.0 -2.0 39.6 -1.6
39 13466917 41.8 -2.8 41.2 -2.2
40 20996011 43.1 -3.1 42.4 -2.4
41 24036583 43.4 -2.4 42.7 -1.7
42 25964951 43.6 -1.6 42.9 -0.9
43* 30402457 44.1 -1.1 43.3 -0.3
44* 32582657 44.3 -0.3 43.5 0.5
45* 37156667 44.6 0.4 43.9 1.1
46* 42643801 45.0 1.0 44.2 1.8
47* 43112609 45.1 1.9 44.2 2.8
48* 57885161 45.9 2.1 45.0 3.0
Note. Our conjecture numbers：
Our conjecture’s deviation：
L&P’s conjecture numbers： L&P’s conjecture deviation：
* It is uncertain that whether any undiscovered Mersenne primes exist between the 42nd （M25，964，951） and the 48th （M57，885，161） in this Table. Therefore the order is provisional.
Theory number of our conjecture in this paper is very similar to the actual number. There are 25 numbers with deviation less than 1， but only one number of deviation greater than 3 where p is equal to 20，996，011. The first 42 numbers are determined but the last 6 numbers are undetermined. For the first 38 Mersenne primes， the maximum deviation of our conjecture is smaller than Lenstra and Pomerance’s conjecture. If there is one or more other Mersenne primes when 25，964，951
The number of Mersenne primes is
， when 5，000≤p≤50，000. The actual number is seven . The number in this paper comes closer to the actual number than Shanks’ conjecture.
Since the approximate number of Mersenne prime is
n=21og2N-log2log2N-1， then 2log2N=n+1+log2log2N， hence . According to， p of the nth Mersenne prime is . Actual values of p are compared with the values in our conjecture and Brillhart’s conjecture showed in Table 2.
Table 2
Values of p for n-th Mersenne Primes
Number The exponent p Our conjecture Our conjecture’s ratio Brillhart’s conjecture value Brillhart’s conjecture Ratio
1 2 2 1.0 1.5 1.3
2 3 3 1.0 2 1.5 3 5 6 0.8 3 1.7
4 7 9 0.8 5 1.4
5 13 14 0.9 8 1.6
6 17 21 0.8 11 1.5
7 19 32 0.6 17 1.1
8 31 48 0.6 26 1.2
9 61 72 0.8 38 1.6
10 89 106 0.8 58 1.5
11 107 157 0.7 86 1.2
12 127 231 0.5 130 1.0
13 521 339 1.5 195 2.7
14 607 496 1.2 292 2.1
15 1279 724 1.8 438 2.9
16 2203 1056 2.1 657 3.4
17 2281 1536 1.5 985 2.3
18 3217 2232 1.4 1478 2.2
19 4253 3238 1.3 2217 1.9
20 4423 4693 0.9 3325 1.3
21 9689 6792 1.4 4988 1.9
22 9941 9822 1.0 7482 1.3
23 11213 14189 0.8 11223 1.0
24 19937 20480 1.0 16834 1.2
25 21701 29537 0.7 25251 0.9
26 23209 42566 0.5 37877 0.6
27 44497 61303 0.7 56815 0.8
28 86243 88231 1.0 85223 1.0
29 110503 126910 0.9 127834 0.9
30 132049 182445 0.7 191751 0.7
31 216091 262144 0.8 287627 0.8
32 756839 376476 2.0 431440 1.8
33 859433 540424 1.6 647160 1.3
34 1257787 775432 1.6 970740 1.3
35 1398269 1112183 1.3 1456110 1.0
36 2976221 1594560 1.9 2184164 1.4
37 3021377 2285318 1.3 3276247 0.9
38 6972593 3274178 2.1 4914370 1.4
39 13466917 4689375 2.9 7371555 1.8
40 20996011 6714162 3.1 11057332 1.9
41 24036583 9610358 2.5 16585998 1.4
42 25964951 13751945 1.9 24878998 1.0
43* 30402457 19673030 1.5 37318497 0.8
44* 32582657 28136247 1.2 55977745 0.6
45* 37156667 40230350 0.9 83966617 0.4
46* 42643801 57509400 0.7 125949930 0.3
47* 43112609 82191237 0.5 188924890 0.2
48* 57885161 117440510 0.5 283387330 0.2
Note. Our conjecture： Our conjecture ratio ：
Brillhart’s conjecture value ： （1.5）n Brillhart’s conjecture ratio ：
Values in this paper and Brillhart’s conjecture have a certain gap with the actual values. Conjecture values are not more than the twice of actual values most of the time. Actual values are most 2.1 times of the conjecture values in this paper when p ≤ 6，972，593. Actual values are right on most 3.4 times of the Brillhart’s conjecture values. The first 42 ratios have been determined and the last 6 ratio are undetermined. If there is no other Mersenne prime when p < 57，885，161， Brillhart’s conjecture values are at most 5 times than actual values for the last 10 Mersenne primes. Nonetheless， actual values are at most 3.1 times than conjecture values in this paper. If there is one or more other Mersenne primes when p < 57，885，161， Brillhart’s conjecture values are at most 8 times than actual values for the last 10 Mersenne primes. But actual values are still at most 3.1 times than conjecture values in this paper. From the above deducing， we can gauss that there are 2 Mersenne primes in average when x ≤ p ≤ 2x where x is large. Actually， sometimes， there is no Mersenne prime between x and 2x. But sometimes there are more than one Mersenne prime. For example， seven Mersenne primes exist at least when x is 24，036，583. When x is smaller， the average of Mersenne primes number is less than 2 between x and 2x.
The probability that MN is a Mersenne prime is
.
So the probability that Mp is a Mersenne prime is
.
4. SUMMARY
It is difficult to find the pattern of Mersenne primes because the distribution of Mersenne primes is very irregular. Based on Zhou’s conjecture， this paper uses formula-transformation method， and derives the approximated expression of Mersenne primes. The numbers of Mersenne primes estimated by this expression are much nearer to the actual numbers than the numbers estimated by Lenstra and Pomerance. The position of each Mersenne prime is also of the view by this expression and the results are better than the estimation in Brillharts’ conjecture. The results in this paper are closer to the actual value than Shanks and Gillies’ conjecture.
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