|15 Indonesian Journal of Mathematics Education Vol. 3, No. 1, April 2020, pp: 15~22 p-ISSN: 2654-3907, e-ISSN: 2654-346X, DOI: 10.31002/ijome.v3i1.2282 e-mail: [email protected], website: jurnal.untidar.ac.id/index.php/ijome

Some New Notes on Mersenne Primes and Perfect Numbers

Leomarich F. Casinillo Visayas State University, Visca Baybay City, Leyte, Philippines e-mail: [email protected]

Received: 19th March 2020 Revised: 27th April 2020 Accepted: 29th April 2020

Abstract Mersenne primes are a specific type of prime number that can be derived using the

formula , where is a prime number. A perfect number is a positive integer of the form ( ) ( ) where is a Mersenne prime and can be written as the sum of its proper divisor, that is, a number which is half the sum of all of its positive divisor. In this paper, some concepts relating to Mersenne primes and perfect numbers were revisited. Mersenne primes and perfect numbers were evaluated using triangular numbers. Further, this paper discussed how to partition perfect numbers into odd cubes for odd prime The formula that partition perfect numbers in terms of its proper divisors were developed. The results of this study are useful to understand the mathematical structures of Mersenne primes and perfect numbers. Keywords: Mersenne primes, perfect numbers, triangular numbers

INTRODUCTION Furthermore, if is perfect, then ( ) . According to Euclid’s proposition If is prime, say , then ( ) . (Niven & Zuckerman, 1980), if as many If is a prime power, say , then numbers as we please beginning from a unit be ( ) . If set out continuously in double proportion until the sum of all becomes a prime, and if the sum is the product of two distinct primes, say multiplied into the last make some numbers, , then ( ) the product will be perfect. In this context, ( )( ). Hence, ()n is a multiplicative double proportion means that each number is function, which is to say, ( ) ( ) ( ). twice the preceding number, as in Niven and Zuckerman (1980) stated For example, is prime; that the sum of proper divisors of a positive therefore, is a perfect number. integer gives various other kinds of numbers. Put simply, if , Numbers in which the sum is less than the where is prime, then is a perfect number itself are called deficient, and that the number. We can use the fact that sum is greater than the number are called to rewrite Euclid’s abundant. A semi-perfect number is a natural results in a modern form. A theorem states that number that is equal to the sum of all or some every perfect number is in the form of of its proper divisors. A semi-perfect number ( ) where is a Mersenne that is equal to the sum of all its proper prime (Ore, 1948). In order to prove this divisors is a perfect number. Most abundant theorem, a helpful function ( ) where is a numbers are also semi-perfect; abundant positive integer, is used to analyze perfect numbers which are not semi-perfect are called numbers (Erickson & Vazzana, 2008). Let weird numbers (Dickson, 1971). These terms, ( ) be the sum of all the positive divisors of together with perfect itself, come from Greek Thus,  nd   , where is a divisor of numerology. In solving a perfect number, there dn|

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is a one-to-one correspondence between intended to expose the new structures of perfect number and of a prime number of the Mersenne primes, perfect numbers, and form . These prime integers are called triangular numbers. Specifically, it aimed to Mersenne primes (Rosen, 1993). develop new claims relating to the said Generally, numbers of the form numbers. We also partitioned perfect numbers

without primarily requirement into odd cubes and derived a formula for odd conditions are called Mersenne numbers. prime Further, we construct a formula on Mersenne numbers are sometimes defined to how to partition perfect numbers in terms of its have the additional requirement that must be proper divisors and determined the number of prime, equivalently that they called pernicious primes in the partition. Mersenne numbers, namely those numbers whose binary representation contains a prime METHOD number of ones and no zeros. The smallest This study is exploratory in nature. composite pernicious Mersenne number is The formula that generates Mersenne primes Dickson (1971) and perfect numbers were presented using the stated that the story of Mersenne numbers concept of Euclid’s proposition. Furthermore, started in the 16th Century, with the French different useful functions related to perfect Monk, Father Marin Mersenne (1588 – 1648). numbers in proving theorems were considered Mersenne had gained an interest in the for the analysis in partitioning these numbers. numbers of the form – 1 (mainly from Mersenne primes and perfect numbers were Fermat’s new tools, like his Little Theorem), evaluated according to its structures, existence, and in 1644 produced Cogitata Physica- and characteristics. Also, those numbers were Mathematica, in which Mersenne stated that evaluated by a triangular number. Calculations is prime for the following values of p: of perfect numbers were then developed, after 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257 and which partitioning these numbers to its proper composite for all other values of p < 257. It divisors were done, and then it leads to the was clear to Mersenne’s peers that he could partitioning formula. Figure 1 presents the not have possibly tested all these values, and as schematic diagram of the flow of the study. it is, his assertion was incorrect. Not every prime value of p in results in a prime, but the chances of being prime are much greater than for a randomly selected number. It took some 300 years before the details of this assertion could be checked completely, with the following outcome: is not a prime for p  67 and p  257 , and is a prime for pp61, 89 and 107. Mersenne made 5 mistakes. Thus, there are 12 primes p  257 such that is a prime. The triangular number is the number of dots in the triangular array with rows that has dots in the th row (Rosen, 1993). For instance,

, and This is nn( 1) defined as Tn, {1,2,3,...}. n 2 Triangular numbers, in fact, is a family of Figure 1. Schematic Diagram of the Research numbers (Montalbo et al., 2015). This study Flow

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RESULTS AND DISCUSSION ( ) ( ) ( ) Note that From the definition of Mersenne prime ( ) and ( ) since above, the following Remark is immediate. we are assuming that is prime. Thus, Remark 1. If is prime, then it is a ( ) ( ) demonstrating that Mersenne prime. is a perfect number.

The next Theorem determines all even Suppose we have a sequence that perfect numbers stated by Euclid and later satisfies a certain recurrence relation and initial developed by Euler into modern form (Ore, conditions. It is often helpful to know an 1948; Rosen, 1993). explicit formula for the sequence, especially if we need to compute terms with very large Theorem 2. (Euclid-Euler) The positive subscripts or if we need to examine general integer is an even perfect number if and only properties for the sequence. The explicit if ( ) where is an integer formula is called a solution to the recurrence such that and is Mersenne prime. relation. The following result for Mersenne primes and perfect numbers involving the Proof. ( ) Let be even perfect number. concepts of recurrence relation stated as Write , where and are positive Theorem 3, is presented as follows. integers and is odd. Since ( ) then we have ( ) ( ) ( ) ( ) Theorem 3. Let . If is ( ) ( ) (eq’n 1). Since is perfect, a prime for some positive integer then then we have ( ) (eq’n 2). i.) is a Mersenne prime; and

Combining eq’n and shows that ( ii.) is an even ) ( ) (eq’n 3). Since ( perfect number. ) then ( ) Hence, there is an Proof (i). Suppose that is prime for some positive integer . integer such that ( ) Inserting Then, we have this expression for ( ) into eq’n 3 tells us that

( ) and therefore ( ) ( ) (eq’n 4). So, and

Replacing by the expression on the left-hand ( ) side of eq’n 4, we find that . ( ) ( ) (eq’n 5). Continuing the process, we obtain Next, we will show that Note that if

then there are at least three distinct ( ) positive divisors of namely and This implies that ( ) which . contradicts eq’n 5. Hence, and from Since , it follows that, eq’n 4, we conclude that Also, from eq’n 5, we see that ( ) , so that

must be prime since its only positive divisors n1 ( ) i are 1 and Thus, where 2 . is prime. i0 ( ) We show that if ( ) (Consider this as equation 1) where is Mersenne prime, then is Multiplying equation 1 with , we have n1 perfect. Note that since is odd, we have 2a   2i1 . n  ( ) Since is a multiplicative i0 function, it follows that (Consider this as equation 2)

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Combining equation 1 and equation 2, it n n11 n i i1 i n   ( i  1)  1  follows that Sn (2  2 )   2  2 )     2  i1 i  1   i  1  1  n1 n  1 n  1 i i11 i i nn11 aann2  2   2   (2  2 ). 2i  2 n  2 i i0 i  0 i  0  ii10 n1 nn11 ii1 i n i 0 Thus, an (2 2 ) 2  2  2  2 i0 ii11 ( ) ( ) nn11 2i  2 n  2 i  1  ( ) ( ). ii11 n By telescoping we obtain, Sn 2 1.

Since is a prime, then by Remark 1, is a Mersenne prime. Since is prime, it follows that Proof (ii). Since is a Mersenne prime for is a Mersenne prime by Remark some positive integer , then it follows 1. that is a prime number by definition. Thus,

Proof (ii). Suppose that is prime for some is an even perfect number by positive integer . Then, is a Mersenne Theorem 2. prime and it follows that is a prime number by definition. Thus, by Theorem 2, Theorem 6. (Dickson, 1971) If is prime and

is an even perfect number. is a positive integer with ( ) , then ( ) Lemma 4. (Leithold, 1996) Let and For our next result involving Mersenne is a function. Then, prime, this is obtained using the Fermat’s last b b c F( i ) F ( i c ). theorem above. This Theorem shows that if a i a i  a  c Mersenne prime is subtracted by 1, then it is The next result on Mersenne primes divisible by whenever is a prime number. and perfect numbers involves sigma notation that concern with the sums of many terms and Theorem 7. Let be an odd prime and be it is a direct consequence of Lemma 4 above. a Mersenne prime. Then, less by This sigma notation is to facilitate writing one is divisible by these sums and use of the symbol ∑ n Proof. Since is an odd prime, then ii1 Theorem 5. If Sn (2 2 ) is a prime ( ) Then, it follows that i1 ( )by Theorem 6. This implies that for some positive integer then ( ) and hence ( i.) is a Mersenne prime; and ) Then, we have ( ) and it implies that ii.) is an even perfect

number. ( ) Thus, ( ) is divisible by n ii1 Proof (i). Suppose that Sn (2 2 ) is a i1 Theorem 8. (Muche et. al., 2017) Let ( ) prime for some positive integer . Then, ( ) be a perfect number. Then, ( ) by Lemma 4, we have is a triangular number.

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Proof. Consider that ( ) ( ) is Thus, by definition of a triangular number, a perfect number. Then, then it follows that ( ) is a triangular ( ) ( ) number.

( ) Theorem 11. Let ( ) and ( )( ) . ( ) ( ). If is

Let be a Mersenne prime. Then, Mersenne prime, then ( ) is a perfect ( ) ( ) and by definition of number. triangular number, ( ) is a triangular Proof. Suppose that ( ) and number. is Mersenne prime. Then, we obtained,

( ( ) ) Since a perfect number is a triangular number by Theorem 8, then a perfect number (( ( ) ) can be partitioned as the sum of natural ( ( ) ) ) numbers from 1 to where is prime. ( ) ( ) This is shown in Remark 9 below. ( ) Remark 9. If is a Mersenne prime, ( ) ( ) 21p  then P() p i is a perfect number.  ( ( ) ) i1 The following results below are the ( ) direct consequences of the definition of a triangular number and Theorem 8. The ( ( ) ) ( ). Thus, by Theorem 2, ( ) is a perfect number. function ( ) in Theorem 10 and 11 is several ways of putting a dominating set in path graph when the order of graph is Lemma 12. (Santos, 1977) The sum of the ( ) (Casinillo, 2018). first terms of an arithmetic progression whose general term ( ) is

( ), where is a common Theorem 10. If ( ) and ( )

( ) then ( ) is a triangular difference. number. We need the Lemma above to prove Proof. Suppose that ( ). Then, the next Theorem. This Theorem involves a for all positive integer . So, we recurrence relation as a function of , where have, is a Mersenne prime.

( ) (( ) ( ) Theorem 13. Let , where ) . If is a Mersenne prime then is a

( ) triangular number.

( ) Proof. Let where and

is a Mersenne prime. Then, we have ( )

( )( ) ( ( )) ( ) ( ) , where ( ( )) ( ) .

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Continuing the process, we end up with, Case 2. Let ( ) Then,

( ) ( ) Then, it follows that ( ) ( ) ( ) ( ) and hence, we have ( ) ( ) ( ) ( ) ( ) , since Case 3. Let ( ) Then, there exists Now, let be a Mersenne prime, a positive integer such that then we obtained So, we have

(( ) ) ( ) ( )( ) (( ) ) (( ) )

( ) By Lemma 12, it follows that ( ) ( ) ( )

( ) (( ) )]

( ) Obviously, it follows that ( ) By Theorem 2, ( ) is a ( ) ( ) Thus, we perfect number. Clearly, we have obtained ( ) ( )

( ) is a triangular number by Theorem 8. This Lemma 17. (Santos, 1977) Let be a natural completes the proof showing that is a n triangular number. number. Then, (2i 1)3  n2(2n2 1).  i1 The following Remark is a direct For the next result that involves how to consequence of Theorem 8 and Theorem 13. partition an even perfect number into odd

cubes, we need the following Lemma below to Remark 14. Let is an odd prime. Then, prove this result. ( ) is a perfect number. ( )

Lemma 15. (Gallian, 2010) Let and be Theorem 18. Let ( ) be a perfect number an integer. If and is relatively prime to where is an odd prime. Then, , then 2( p1) / 2 P( p)  (2r 1)3. The next result for a perfect number is  r1 obtained using the concepts of congruencies Proof. Suppose is an odd prime. Then, it and the generalized Euclid’s lemma above. must be shown that 2( p1) / 2 Theorem 16. Let ( ) ( ) be a (2r 1)3  2 p1(2 p 1). So, by Lemma perfect number. Then, one of the following r 1 two conditions hold: 17, we obtain 2 2 i.) ( ) ( ) or 2( p1) / 2  p1    p1   (2r 1)3  2 2  22 2  1 ii.) ( ) ( )      r 1         ( ) p1 p1 Proof. Let Then, ( ) is  2 [2(2 ) 1] p1 p a triangular number by Theorem 8.  2 (2 1). Then, consider the following cases: This completes the proof showing that each even perfect number ( ) ( ), Case 1. Let ( ). Then, ( ) ( ) and so we obtained where is an odd prime can be written as the ( ) ( ) by Lemma 15. sum of odd cubes.

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As a direct consequence of Theorem formula that partitions a perfect number into 18, Remark 19 is obtained and presented its proper divisors. below. Theorem 23. Let ( ) ( ) be a ( )⁄ Remark 19. There are distinct odd perfect number. Then, the formula for cube summands in a perfect number of the partitioning an even perfect number in terms of form ( ) ( ), where is an odd proper divisors is given by prime. pp1 Pp( ) 2i11  2 i (2 p  1) . From the definition of perfect numbers ii11 above, Remark 20 and Theorem 21 are Proof. Suppose that ( ) ( ) is a obtained. The following result shows that perfect number. Then, we have perfect numbers are half the sum of all of its positive divisors. ( ) ( ) ( )] Remark 20. Let be a perfect number. If ( ) ( ) P then  2.P ( ) ( )   dP| d ( ) Continuing the process, we obtained Theorem 21. If is a perfect number and ( ) ( ) ( ) is a positive integer, then ( ) ( ) 1 2P  1  . ( ) ( ) ( )  dP dP|  ( ) ( ) Applying Lemma 22, we have Proof. Suppose that is a perfect number and ( ) ( ) is a positive integer. If , then it ( ) ( ) implies that is a proper divisor of Note that if then by definition, ( ). Clearly, we end up with, P  ()Pd p p1 d d|| P d P  P( p)  2i1  2i1(2 p 1) .   By Remark 20, we obtained, i1 i1 P  1   PP   2. It follows that Definition 24. (Voight, 1998) The number of dd    d|| P d P proper divisors of a perfect number is the 1 11 number of terms in the partitioned perfect  2. But so 2. dP| d dP| dP number and it is denoted by ( ) 1 2P  1 The next Theorem is immediate from Thus,   . dP| dP Theorem 23 and Definition 24.

Lemma 22. (Leithold, 1996) If Theorem 25. The number of proper divisors of is a geometric sequence with a perfect number of the form ( ) common ratio , and ( ) is given by ( ( ))

( ) then Proof. Let ( ) ( ) be a perfect It is worth noting that in partitioning a number. Then, we have p p1 perfect number , is written as a sum of its P( p)  2i1  2i1(2 p 1) by Theorem proper divisors. Hence, Lemma 22 below, i1 i1 Theorem 23 is obtained that presents the 23.

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By Definition 24, it follows that, Dickson, L. E. (1971). History of the theory of p p1 numbers. New York: Chelsca Pub. Co.  i1 i1 p  D(P( p))  D2  2 (2 1) .  i1 i1  Erickson, M. & Vazzana, A. (2008). Clearly, we have ( ( )) Introduction to number theory. New York: Taylor and Francis Group. Hence, ( ( ))

Gallian, J. A. (2010). Contemporary abstract The following Remarks are direct algebra. (7th ed.). USA: Brook/Cole, consequences of Theorem 23 and 25. These Belmont. determine the number of composite and prime Leithold, L. (1996). The calculus 7. (7th ed.), numbers in the summands of a partitioned Addison-Wesley Publishing Company, perfect number. Inc.

Remark 26. Let ( ) ( ) be a Montalbo, J., Berana, P. J., & Magpantay, D. perfect number. Then, ( ) has (2015). On triangular and trapezoidal composite proper divisors. numbers. Asia Pacific Journal of Multidisciplinary Research, 3(1), 76-81.

Remark 27. Let ( ) ( ) be a Muche, T., Lemma, M., Tessema G., & Atena, perfect number. Then, and are the A. (2017). Perfect if and only if only primes among its proper divisors. triangular. Advances in Theoretical and Applied Mathematics, 12(1), 39-50. CONCLUSION New claims on Mersenne primes, even Niven, I. & Zuckerman, H. S. (1980). An introduction to the theory of numbers. (4th perfect numbers and triangular were obtained ed.). Canada: John Wiley and Sons, Inc. relating to recurrence relation and summation notations. The result shows that a Mersenne Ore, O. (1948). Number theory and its history. prime of the form less by one is New York: McGraw-Hill. divisible by The study also shows that all Rosen, K. H. (1993). Elementary number even perfect numbers are triangular numbers, rd theory and its application (3 ed.). USA: thus, the form ( ) ( ) can be Addison-Wesley Publishing Company. partitioned as the sum of the integers from 1 to

Mersenne prime of the form . Santos, E. Z. D. (1977). Advanced algebra. Further, the study shows that a perfect number Manila: Pioneer Printing Press. can be written as the sum of ( )⁄ odd Voight, J. (1998). Perfect numbers: An cubes and ( ) ( ) or ( ) elementary introduction. Berkeley: ( ). Also, perfect numbers can be Department of Mathematics, University partitioned as a sum of proper divisors of California. with composite proper divisors and two prime divisors namely: even prime number 2 and Mersenne prime where is a prime number.

REFERENCES Casinillo, L. F. (2018). A note on Fibonacci and Lucas number of domination in path. Electronic Journal of Graph Theory and Applications, 6(2), 317–325. doi:10.5614/ejgta.2018.6.2.11

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