1 Introduction. n = Key-Words: - Mersenne numbers, prime numbers, Generalized Mersenne numbers, distributions

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1 Generalized Mersenne prime numbers: characterization and distributions VLADIMIR PLETSER Microgravity Projects Div., European Space Research Technology Centre, European Space Agency Dept MSM-GMM, P.O. Box 2, NL-22 AG Noordwijk, THE NETHERLANDS Abstract: - Mersenne numbers are a particular case of a larger class of numbers, the Generalized Mersenne numbers [a n - (a - 1) n ], characterized by their base a and exponent n. Besides known Mersenne primes, other primes can be generated with same prime exponents n using the generalized Mersenne formulation. In addition, for prime exponents n that do not yield Mersenne primes, Generalized Mersenne primes can also be generated. The distributions, densities and properties of these primes are investigated. These distributions follow power laws similar to the one of the distribution of natural primes. Key-Words: - Mersenne numbers, prime numbers, Generalized Mersenne numbers, distributions 1 Introduction If a Mersenne number M n = (2 n - 1) is prime, then n is prime. The reciprocal is not true, as e.g. for n =, M = 2 (= 23x) is composite. There are 31 Mersenne prime numbers known (for review, see e.g. [1]). The reason why certain prime values of the exponent n yield Mersenne primes and others yield Mersenne composites is not known. Mersenne numbers are a particular case of a larger class of numbers, the Generalized Mersenne numbers GM a,n = [a n - (a - 1) n ] (1) characterized by their base a and exponent n. In this paper, Generalized Mersenne primes and their distributions are studied. These distributions follow power laws of the distribution of natural primes. 2 Generalized Mersenne primes Prime GM a,n were searched for values of a between 2 and 1 for the first ten odd prime values of the exponent n, 3 = n = 31. Among the 1 Generalized Mersenne numbers investigated, only 2 are primes (< 1%), excluding 1. The smallest number is prime, GM 2,3 = M 3 =, the largest is composite, GM 1,31 which has 2 digits. Table 1 shows those values of a yielding GM a,n primes with the corresponding number d of digits. Interestingly, it shows that for prime values of n that do not yield Mersenne primes (i.e. n =, 23 and 2), prime GM a,n can still be found for values of a > 2. Other characteristics can be deduced from these distributions. First, the size of the GM a,n primes is increasing at a moderate rate for a same value of n and increasing values of a, but rapidly for increasing values of n (from d = 1 for GM 2,3 to d = 2 for GM,31 ). Second, the number of GM a,n primes decreases as expected for increasing prime values of n. However, this decreasing is not monotonous. Fig. 1 shows the number of GM a,n primes found for 2 = a =1 for the first eleven prime values of n Fig. 1: Number of GM a,n primes for 2 = a =1 and 2 = n prime = 31 n = Some prime exponents n (n =, 23 and 2) seem less "productive" than others in generating GM a,n primes, as their numbers are less than for adjacent prime values of n. Interestingly, these "less productive" exponents n are also those yielding composite Mersenne numbers.

2 Table 1 : Values of a yielding GM a,n primes and number d of digits for 2 = a = 1 and 2 = n = 31 n = 3 n = 5 n = n = n = 13 n = 1 n = 1 n = 23 n = 2 n = 31 d a d a d a d a d a d a d a d a d a d a

3 3 Distribution of GM a,n primes in function of a The distribution of GM a,n primes in function of a for each n is not homogeneous. For 1 = a = 1 and n prime = 31, Fig. 2 shows the density of GM a,n primes (indicated by dots) and their numbers in bins of hundreds of the base a. bin n = 3 n = 5 n = n = n = 13 n = 1 n = 1 n = 23 n = 2 n = a Fig. 2: Numbers of GM a,n primes in bins of hundreds of a for 1 = a = 1, 3 = n prime = 31. Among the distributions for the more "productive" exponents n (i.e. n = 3, 5,, 13, 1, 1, 31), the first one (n = 3) is relatively regular and homogeneous with a slowly decreasing density. For the following ones, gaps of increasing sizes for increasing n appear between primes. These distributions, although having their own peculiarities, can be characterized by positions in the various bins of peak and trough values. Three categories are seen : 1) those distributions for n = 3, 1, 1, with the highest number of GM a,n primes in the first bin, with a second peak in bins 5 or, separated by 3 or bins with smaller values ("large valley" type), and with a third peak in the th or 1th bin (n = 3, 1) or in the th bin (n =1); 2) those distributions for n = 5,, 31, with the first peak in the first bin, with a second peak in the 3rd or th bin, separated by 1 or 2 bins with less primes ("steep valley" type), with a third peak in bin (n =, 31) or (n = 5), followed by a fourth peak in bins or 1 (n =, 31); 3) the distribution for n = 13, with the first peak in the 3rd bin and three other peaks in bins 5,, 1. Among the distributions for "less productive" exponents, the one for n = has a first peak in bin 1 and a second peak in bin, followed by slowly decreasing values. Note also the large gaps (bins 2, 5,,, ) and regions where primes are more evenly distributed (bin 3 and part of bin, and bin and part of bin ). For n = 23, except for the small peak in the first bin, the distribution is constant if bins 5 and are combined. The distribution for n = 2 is the strangest, characterized by four empty bins (2,,, ) and by a nearly perfect symmetry around the middle of bin 5 (if bins and 1 are combined). Furthermore, some of the GM a,n primes appear for successive values of a. Table 2 shows the number of such pairs [a, (a+1)] and the number of interwoven pairs [a, (a+2)], including some triplets and quadruplets (counted respectively as 2 and 3 pairs), for which the corresponding GM a,n are prime. Relative values (number of pairs divided by total number of GM a,n primes for each n) are also indicated. Again it shows minima for the "less productive" exponents n =, 23 and 2. The numbers of pairs and interwoven pairs are decreasing for increasing n, but not monotonously. Table 2 : Numbers of pairs [a, (a+1)] (a) and interwoven pairs [a, (a+2)] (b) yielding prime GM a,n n a) % b) % First GM a,n primes for n prime < 2 The "productivity" of exponents n to generate GM a,n primes can be assessed also by the bases a values for which the second (or first) GM a,n primes occur for prime exponents n yielding Mersenne primes (or composites). Table 3 shows the values of a > 2 for which the second (or first) GM a,n primes with d digits occur for prime exponents n < 2 yielding respectively Mersenne primes (or composites). Among the exponents n yielding Mersenne primes, the second GM a,1 prime for n = 1 appears surprisingly late for a = 15. One can suspect that, although n = 1 yields a Mersenne prime, the distribution of GM a,1 primes would not be as dense as those for adjacent prime values of n. Among the exponents n yielding Mersenne composites, we know already that the first three exponents are "less productive" than those yielding Mersenne primes, although the first GM a,n primes appear relatively early for a > 2. The following exponents n are also expected to be "less productive" in generating GM a,n primes for

4 those values of n for which the first GM a,n prime occurs for large values of a, say e.g. a =. The difficulty of computing GM a,n prime distributions with these exponents n is obvious, looking at the size of the corresponding first GM a,n primes, e.g. for n = 13, the first GM a,n prime appears for a = 32 and has 35 digits. Table 3 : For prime n < 2, values of a > 2 yielding GM a,n primes with d digits M n primes M n composites n a d n a d n a d n a d Density and distribution of GM a,n primes in function of n Similarly to the function?(n) giving the number of natural primes less than an integer N, one defines the functions? GM (N, n) giving the number of GM a,n primes less than N and generated with the prime exponent n. As the GM a,2 primes yield all the natural primes but the single even prime 2, one has? GM (N, 2) =? (N) - 1 for N > 2 (2) The density of GM a,n primes is defined as? GM (N, n) divided by N. As the distribution of natural primes is a function of N, that can be represented in first approximation by [N / Log(N)], the distribution of GM a,2 primes is in first approximation? GM (N, 2)?(N) [N / Log(N)] (3) where Log is the Napierian logarithm and where the approximation is growing better for larger N. For n > 2, we ask whether other distributions of GM a,n primes follow also [N / Log(N)] laws to powers P n, or in general? GM (N, n) = K n [N / Log(N)] Pn () where K n and P n are constants, different for each n. Table gives in function of the number d of digits, the values of? GM (N, n) for values of N in powers of 1 (N = 1 d ) and for n = 2, 3, 5 and. Data for n = 2 are from [1, 2]. Data for n = 3 are from Table 1 for 2 = a = 1, and from an additional computation for 11 = a = yielding 2 and 1 GM a,3 primes having respectively and digits. Data for n = 5 and are from Table 1 with 2 = a = 1. Table : Distribution of GM a,n primes n = 2 n = 3 n = 5 n = d? GM? GM e d ε d? GM e d ε d? GM e d ε d The parameters e d and ε d are defined as follows. As? GM (N, n) becomes smaller for increasing n, the exponent P n is smaller than 1 and should be a decreasing function of n. Writing P n = 1/E n < 1 (5) and if relation () holds, the values of E n can be estimated in two different ways. First, by considering the ratio of? GM (N, n) for two successive values of d, i.e. for N increasing by one order of magnitude from N d to N d+1, one obtains from () the first estimates e d of E n Log (N d / Log N d ) - Log (N d+1 / Log N d+1 ) e d = () Log [? GM (N d ) /? GM (N d+1 )] Second, by taking the? GM (N, 2) as the "mother" distribution in its first approximation (3), the other distributions? GM (N, n) in their form () can be seen

5 as the E n -th root of? GM (N, 2). Whether the distribution?(n) of natural primes should be taken instead of? GM (N, 2) as the "mother" distribution is an academic question that we leave aside as we are only interested in the distribution values and both distributions are close enough for this purpose. The second estimates of E n are the exponents ε d computed for each value of d such as the e d -th root of? GM (N d, 2) yields the value of [? GM (N d, n)] or ε d = Log [? GM (N d, 2)] / Log [? GM (N d, n)] () Note that in Table, indicative values of ε d for n = and d > 1 were obtained by replacing? GM (N d, 2) by (N d / Log N d ) from (3) as values for?(n) are not available for d > 1 (see e.g. [1]). Fig. 3 shows the behaviour of the sequences of e d and ε d : starting from small values for e d and large values for ε d, both series oscillate around an average value to eventually converge toward values of E n approximately 2, 5 and respectively for n = 3, 5 and. Furthermore, if? GM (N, n) is the E n -th root of? GM (N, 2), then the constants K n should also take the value 1 for N large enough d 2 Fig. 2 : Plots of e d (black symbols) and ε d (white symbols) as estimates of E n in function of d (see text) for n = 3 (thick interrupted lines with squares), n = 5 (continuous lines with diamonds) and n = (thick interrupted lines with circles). Mersenne primes, Generalized Mersenne primes can also be generated. The numbers of GM a,n primes less than a certain integer and generated with different prime exponents n are on average decreasing for increasing n. However, it is found that some exponents n are "less productive" than adjacent prime values of n in producing GM a,n primes, their density distribution being less than for other adjacent prime values of n. Furthermore, GM a,n primes are not homogeneously distributed with respect to the base a. As could be expected, these "less productive" exponents n are those producing Mersenne composites. However, it is suspected that all prime exponents n yielding Mersenne primes would not be as "productive" as others, as the second GM a,n prime for n = 1 occurs for a value of the base a relatively large. In this respect, one should not look upon prime values of n yielding Mersenne primes as having anything special that other prime values of n yielding Mersenne composites would not have. Instead, using the Generalized Mersenne formulation (1), it appears that some prime exponents n and some ranges of bases a are "more productive" than others in generating GM a,n primes, including Mersenne primes (which are simply GM a,n primes for a = 2). Regarding the GM a,n prime distributions, it is found that they follow E n -th root power laws of the natural prime distribution. The GM a,n prime distribution for n = 2 is very close to the one of the natural primes (E n 1), while the one for n = 3 is close to the square root of the natural prime distribution (E n 2). For other values of n, the exponents (1/E n ) decrease with increasing n. References: [1] P. Ribenboim, The book of prime number records, Springer-Verlag, New-York, 1, 5-1. [2] P.J. Davis and R. Hersh, The mathematical experience, Houghton Mifflin Company, Boston, 11, Conclusions Besides the known Mersenne primes, one can generate other primes with the same prime exponent n by using the generalized Mersenne formulation (1). In addition, for prime values of the exponent n that do not yield

ALIQUOT SEQUENCES WITH SMALL STARTING VALUES. Wieb Bosma Department of Mathematics, Radboud University, Nijmegen, the Netherlands

#A75 INTEGERS 18 (218) ALIQUOT SEQUENCES WITH SMALL STARTING VALUES Wieb Bosma Department of Mathematics, Radboud University, Nijmegen, the Netherlands bosma@math.ru.nl Received: 3/8/17, Accepted: 6/4/18,