SIEVE METHODS FOR ODD PERFECT NUMBERS

Transcription

1 SIEVE METHODS FOR ODD PERFECT NUMBERS S. ADAM FLETCHER, PACE P. NIELSEN, AND PASCAL OCHEM Abstract. Using a new factor chain argument, we show that 5 does not divide an odd erfect number indivisible by a sixth ower. Alying sieve techniques, we also find an uer bound on the smallest rime divisor. Putting this together we rove that an odd erfect number must be divisible by the sixth ower of a rime or its smallest rime factor lies in the range 0 8 < < These results are generalized to much broader situations.. Introduction A ositive integer of the form 2 n is called a Mersenne number. These numbers can only be rime when n is rime. As it currently stands, there are forty-seven known Mersenne rimes. The largest is , which is also currently the largest exlicitly known rime, and the first such found with more than ten-million digits. As observed by Euclid, a number of the form N = 2 2 ), where 2 is a Mersenne rime, ossesses the roerty that σn) = 2N, and Euler roved the converse in the case when N is even. A ositive integer N satisfying σn) = 2N is called a erfect number. There are two questions, oen since antiquity, concerning erfect numbers. Question. Are there infinitely many even erfect numbers? Equivalently, one can ask if there are infinitely many Mersenne rimes. Due to heuristic arguments, and the fact that rime-hunters kee finding more of them at about the right times, the widely held belief is that the answer to this question is yes. However, little concrete rogress has been made in formally answering the question. A ositive answer would give an answer to other difficult oen questions, such as whether or not the quadratic olynomial 2x 2 is rime infinitely often. Those wishing to look for Mersenne rimes can articiate in GIMPS the Great Internet Mersenne Prime Search) by downloading a free rogram that runs on sare comuter cycles from htt:// Question 2. Are there any odd erfect numbers? The oular oinion is that the answer is no. Further, there are a number of known limiting conditions. We give an udated version of the list rovided in [24]. Write N = k i= ai i where each i is rime, < 2 <... < k, and k = ωn) is the number of distinct rime factors. Then: Eulerian Form: We have N = π α m 2 for some integers π, α, m Z >0, π m, with π α mod 4) and π rime. The rime π is called the secial rime of N. Lower Bound: Brent, Cohen, and te Riele [2] using a comuter search found that N > As of this writing, the third named author has comleted the calculations which imrove this bound to N > 0 500, and further rogress is likely. William Li is heling coordinate this effort via his website htt:// Uer Bound: Dickson [6] roved that there are finitely many odd erfect numbers with a fixed number of distinct rime factors. Pomerance [26] gave an effective bound in terms of k. This was imroved in succession by Heath-Brown [3], Cook [5], and finally by the second author to 200 Mathematics Subject Classification. Primary A25, Secondary N36,A5, Y99. Key words and hrases. abundance, factor chains, large sieve, odd erfect number.

2 2 S. ADAM FLETCHER, PACE P. NIELSEN, AND PASCAL OCHEM N < 2 4k. Recently Pollack [25] used these bounds to give an effective bound on the number of odd erfect numbers in terms of k. Large Factors: Goto and Ohno [9] roved that k > 0 8, and Iannucci [4, 5] roved k > 0 4 and k 2 > 0 2. Small Factors: The smallest rime factor satisfies < 2 3k + 2 as roved by Grün []. For 2 i 6, Kishore [20] showed that i < 2 2i k i + ), and this has been slightly imroved by Cohen and Sorli [3]. Number of Total Prime Factors: Hare [2] roved that the total number of not necessarily distinct) rime factors of N must be at least 75. The third author is erforming comutations which increase this bound to 0. Number of Distinct Prime Factors: The second author [24] demonstrated that ωn) 9 and if 3 N then ωn) 2. In a work in rogress these bounds are increased further. The Exonents: Set d = gcd i a i + ), where the a i run over the exonents on the non-secial rimes. McDaniel [23] roved that d 0 mod 3). If the exonents of the non-secial rimes are all equal to d ), Yamada [30] roved that there is an effective uer bound on N deending only on d. In this case, a number of authors have demonstrated that certain values of d are unaccetable. The focus of this aer will be on the exonents of a urorted odd erfect number, the last of the bulleted oints above. As above, let d = gcd i a i + ), where the a i run over the exonents on the non-secial rimes. With modern comuting ower it is a simle matter to show that d 0 mod 3); taking fewer than 30 cases to check. We even rovide such a roof later in the aer. If we relace 3 by a larger rime, like 5, already the comutation becomes much more difficult, and reviously had never been accomlished. Nonetheless, there is still a straightforward algorithm to deal with this case; assuming you have a large amount of comutational ower at hand. However, that leaves oen what haens in the case d = gcd i a i + ) =. For examle, it is known that the non-secial rimes cannot all have an exonent of 2, nor can they all have exonent 4, but it is unknown what haens in the mixed case when we allow both 2 and 4 to occur. A new idea was develoed by Yamada [29] in a rerint available at the mathematics rerint server htt://arxiv.org since Using sieve methods, he roved the following: Yamada s Theorem. Let n, d, b, b 2,..., b t be ositive integers, where the b j belong to a finite set P. If N = a as s q b qbt t is the rime factorization of N and σn)/n = n/d, then N has a rime factor smaller than a constant C, effectively comutable in terms of n, s, and P. Yamada exlicitly comuted the constant C in the case that N is an odd erfect number i.e. n = 2 and d = ) where the exonents of the non-secial rime factors are either 2 or 4 so s and P = {2, 4}). The constant he obtained was ex ) > 0 20,000,000,000. In this aer we strengthen Yamada s theorem by weakening the condition the b j belong to a finite set to there is a finite set of rimes so that b j + is divisible by one of them. Furthermore, we simlify the roof a great deal, and the comutable constant obtained is much smaller. Alying our theorem, we arrive at the following: Main Theorem. Let N be an odd erfect number such that if a N and is not the secial rime then either 3 a + ) or 5 a + ). The smallest rime factor of N belongs to the range 0 8 < < In Sections 2-4 we develo the machinery necessary to establish an imroved version of Yamada s theorem. Emhasis is given to simlifying the comutations, rather than otimizing constants. Section 5-6 describe how the lower bound of the main theorem is ushed to 0 8. The most difficult roblem, and the real heart of the aer, is dealing with the case when = 5. In Sections 7-0 we finish the roof of the uer bound in the main theorem. The last few sections focus on oen questions and ossible future imrovements.

3 SIEVE METHODS FOR ODD PERFECT NUMBERS 3 2. Break-down of the main ideas We begin with some standard notation. We define the function σ a) = σa) a = d a d for ositive integers a, and note that this function is multilicative. Clearly, an integer N is erfect if and only if σ N) = 2. Given a rime we have σ a ) = a+ a ) which is increasing in a and lim a σ a ) =. An integer N > is said to be n/d-erfect if σ N) = n/d. Reducing if necessary, we will always assume n and d are relatively rime ositive integers. In the case n = 2 and d =, we let π denote the secial rime factor of N. We also let Φ m x) denote the mth cyclotomic olynomial. Hereafter, we fix an integer s 0 and a finite set of rimes P. We will think of the set P as limiting the exonents of all but s of the rime factors of N, but to make this more formal we need further notations. We set P = P and let P max denote the largest rime in P. We let N be an n/d-erfect number with a rime factorization N = s i= r ai i t j= q bj j where for each j t there is a rime j P with j b j + ). In articular we have Φ j q j ) σq bj j ), for any such rime j P. For each P we let Q = {q : q b N and b + )}. Notice that rime factors of N may occur in more than one of the sets Q, but by design) there are at most s rime factors of N which belong to none of them. Without loss of generality we also assume r < r 2 <... < r s. We seak of the rimes q j as the constrained rimes, as their exonents are constrained to secific residue classes. The rimes r i are unconstrained, but limited in number. We will construct a constant, which we call C, that is an uer bound on the smallest rime factor of N. The definition of C is done in three stages; we will assume that C is the smallest constant satisfying the conditions given in each stage. For the first stage we simly assume C max{p max, n} +. This takes care of the case that one of n, d, or P shares a rime factor in common with N. Hereafter, we will assume that N does not share such a factor. Noting that σn) is an integer, we derive from the equation σn) = Nn/d that d N, and hence we may now assume d =. The second stage is also straightforward, and yields a large imrovement over [29]. From the equation σ N) = n we have n = s i= σ r ai i ) j s σ q bj j ) r + i r + i 2 i= P q Q q q = r + s r q q. P q Q q q Q We wish to turn this inequality into a non-trivial lower-bound on the quantity P q. This occurs only when n > r+s r. Equivalently we need r > s+n n. For ease later, we define the following constants: C 0 = s+n n and C 0 = nr ) s + r q q. P q Q Our second assumtion is C C 0 +. The observant reader will note that the constant C 0 deends on more than just n, s, and P; namely the rime r. However, note that with the assumtion C C 0 + in lace we lose no generality by assuming r > C 0. Thus, we also have the inequality C 0 >, and C 0 is bounded away from by a ositive quantity only deending on n, s, and P. The third stage, which is exlained fully in the next two sections, relies on the following idea: We wish to count the number of rimes factors of N which are congruent to mod P ). To this end, fix one of the constrained rimes q Q. If q mod P ) then Φ q) σq b ) σn) = Nn. From our first assumtion on C, we know N, hence n. Thus there are at most v n) rimes q Q with q mod P ), where v denotes the -adic valuation. In articular, there are at most C = s + P v n) distinct

4 4 S. ADAM FLETCHER, PACE P. NIELSEN, AND PASCAL OCHEM rime factors of N which are mod P ). We will use the large sieve to show that this constraint imlies that P q Q q/q ) converges quickly. If this double roduct involves only sufficiently large rimes then we can guarantee it is smaller than C 0, which yields a contradiction. 3. The Large Sieve In this section we fix a rime 0 P and we let reresent an arbitrary rime. To use the large sieve we first need to construct aroriate sieving sets. For mod P ) we set Ω = {0 mod )} {a mod ) : a mod ), a 0 mod )}, and for all other rimes we set Ω = {0 mod )}. We wish to estimate the size of the set S = {n x : n / Ω for all < u}, where x, u R >0. To see why we want to bound the size of S consider the following argument. Assume q Q 0, q mod ), and q 0 mod ). From the definition of Q 0, this imlies that N. If we further assume that C = 0 then there are no rimes N with mod P ). Thus, under the assumtions above, q / Ω unless q =. In articular we have the uer-bound {q Q 0 : q < x} S + u. Since C is not necessarily 0 a small corrective factor needs to be introduced, which we do at the end of this section. If mod P ) we set κ) = 0. Otherwise we set κ) =. We extend κ multilicatively, and define it to be zero on non-square-free arguments. Note that κ) = Ω. We now describe how to give an effective bound on S. Proosition The Large Sieve [6, Theorem 7.4]). We have where Gu) = n u µ2 n) n ) κ) κ). S x + u2 Gu) Note that we are in the situation where small sieves would aly, and this is the avenue Yamada exlored. However, to avoid comlications with error-term estimates we refer the large sieve. In the sieve we take u = x, and thus we need to find a lower-bound for G x). Set V P w)) = <w κ) ). One can aroximate Gu) from above and below by suitable multiles of V P w)), choosing w aroriately in either case. We are concerned with the lower bound, and thus aeal to the following roosition, which follows from a trick of Rankin. Proosition 2 [0, Theorem,. 52]). Suose there is a constant B > so that κ) log) < B logz) <z for some z 2. Writing z = x /s, if s > 2B we have where C 2 s) = s 2 logs/2b) s 2 + B. So we first find an uer-bound of the form logz) G x) e C2s) V P x /s )) <z κ) log) An effective but simlistic uer-bound is easy to achieve. It is well known e.g., see [28, Equation log) 3.24]) that logz) <z <. Since κ) 0 P we can take B = P, and then take s = 2P +. < B.

5 SIEVE METHODS FOR ODD PERFECT NUMBERS 5 Next, we need to find an uer bound for V P w)), where w = x /s. We begin by noting V P w)) ) κ) = ) ) 0 <w <w <w, mod P ) = ) 0 ) 0+. <w <w, mod P ) Thus, it suffices to find an effective uer-bound for the Mertens roduct over the arithmetic rogression mod P ); or lower-bounds for the roducts over the other arithmetic rogressions modulo P. The bounds given in [] suffice. To aly the results of that aer we may need to increase P so we have P 37, which can be done by adding rimes to P if necessary.) We arrive at an effective inequality of the form V P w)) < C 3 logw) + 0 ϕp ) where C 3 > 0 deends only on P, and ϕ is the totient function. At this oint we still need to deal with the ossibility that there might be rime factors of N which are mod P ). Let T denote the set of all such rime factors, if any. We redefine Ω and κ, so that Ω = {0 mod P )} and κ) = whenever T. Notice that we now truly do have the bound {q Q 0 : q < x} S + u. One of the effects of this change in the definition of Ω and κ is that the needed constant B is no larger, so we use the same constant B = P as in our revious comutation. The uer-bound on V P w)) is changed by a constant which deends only on the set T. Further, the worst case is when T consists of the first C rimes which are mod P ). Letting T consist of the first C rimes which are mod P ), we have C 3 V P w)) < logw) + 0 ϕp ) where C 3 = C 3 T 0 is still effectively comutable. Note that at the cost of weaker bounds, we then have > 0 {q Q 0 : q < x} < C 4 x/ logx) +/ϕp ) where C 4 > 0 deends on P, but is indeendent of Partial summation bound We are now ready to estimate q q Q 0 q. Suose that all the rime factors of N are greater than some integer y >. By artial summation we have log q = log /q) = q q + 2q 2 + ) 3q 3 + q Q 0 q Q 0 < y C 4 x logx) +/ϕp ) dx + q>y q Q 0 qq ) < ϕp )C 4 logy) + /ϕp ) nn ) = ϕp )C 4 logy) + /ϕp ) y. n>y

6 6 S. ADAM FLETCHER, PACE P. NIELSEN, AND PASCAL OCHEM We can incororate /y into the main term as the main term dominates), by changing C 4 to a new constant C 5 > 0. Thus 0 < logc 0) log q < P ϕp )C 5 q logy). /ϕp ) Solving for y we have y < ex 0 P q Q 0 P ϕp ) ) ϕp ) )C5 logc 0 ) and our third assumtion on C is that it is larger than the quantity on the right. We have thus roven: Theorem 3. Let N > be an odd integer with σ N) = n/d. Let s be a fixed integer, and let P be a finite set of rimes. Suose the rime factorization of N has the form N = s t i= rai i j= qbj j, where for each j there is a rime j P with j b j + ). There exists a comutable constant C, deending only on P, n, and s, which gives an uer bound on the smallest rime factor of N. 5. Our secial case, dealing with small rimes 5 In the remainder of the aer we secialize to the case when σ N) = 2, P = {3, 5}, and s =. We further assume that the one rime factor of N not necessarily limited by P is the secial rime, π. Note that while the exonent of π is not limited by P we do know that the exonent is mod 4). In articular, Φ 2 π) = π + divides 2N. To abbreviate these assumtions, we refer to an integer N satisfying the above conditions by the acronym R-OPN a restricted odd erfect number). Let N be an R-OPN. Suose for a moment that Q 5 is emty, and so all of the exonents of the non-secial rimes are 2 mod 3). This imlies that Φ 3 q) N for all q N, q π. We wish to show that 3 N when N is an R-OPN, working by contradiction. The next aragrah demonstrates how this is accomlished in one secial case. If 3 N then since π 3 we have 3 = Φ 3 3) N. In other words, starting with the rime 3 we are forced to have 3 as another rime divisor of N. We say that 3 contributes the rime factor 3. Since 3 mod 4) it might be the secial rime, and so we will first consider that case. With π = 3 we have 2 7 = π + = Φ 2 π) 2N. This imlies that 7 Q 3 since 3 is already the secial rime, and we are assuming Q 5 = ) and so 7 contributes the factors 3 9 = Φ 3 7) to N. Reeating this argument we have that 9 contributes 3 27 = Φ 3 9), which in turn contributes the factors = Φ 3 27) N, and finally we have = Φ 3 549). We could continue this rocess of obtaining more and more factors of N, but ausing for a moment to collect our data we currently have M = N. On the other hand σ M) > 2. Adding more rime factors to M, or increasing the exonents on the factors we already have, only increases the size of σ M), which contradicts σ N) = 2. One calls an integer M > for which σ M) > 2 an abundant number. If M N and M is abundant, then N is abundant. Using this fact we were able to show that the situation in the revious aragrah led to a contradiction. However, there are more cases to consider; such as when 3 is not the secial rime. The rint-out below covers all cases. Some exlanation of the notation in the rint-out may be necessary. Every line is of the form a Φ a+ ), where Φ a+ ) is comletely factored to tell us which new rimes are contributed. The exonent a is determined by the case under consideration we have a = if is the secial rime, and a = 2 otherwise). Also note that the letter A tells us that the case under consideration gives us an abundant number, which allows us to back-track to the next case. The indentation tells us how far into a chain of factors we are, and also aids us in finding the next case to consider.

7 SIEVE METHODS FOR ODD PERFECT NUMBERS 7 3^2 => 3^ 3^ => 2^ 7^ 7^2 => 3^ 9^ 9^2 => 3^ 27^ 27^2 => 3^ 549^ 549^2 => 3^ 3^ 33^ 009^ A 3^2 => 3^ 6^ 6^ => 2^ 3^ 3^2 => 3^ 33^ 33^2 => 3^ 7^ 5233^ 7^2 => 3^ 9^ A 6^2 => 3^ 3^ 97^ 97^ => 3^ 7^2 7^2 => 3^ 9^ A 97^2 => 3^ 369^ 369^ => 2^ 5^ 37^ A 369^2 => 3^ ^ ^ => 2^ ^ ^2 => 3^ ^ ^2 => 3^ 6^ 79^ 27^ ^ ^ 79^2 => 3^ 7^2 43^ A ^2 => 3^ ^ ^ => 2^ 5^ 443^ ^ A ^2 => 3^ 8^ ^ ^ 8^ => 2^ 7^ 3^ 7^2 => 3^ 9^ A 8^2 => 3^ 79^ 39^ 79^2 => 3^ 7^2 43^ A Note that the line 33^2 => 3^ 7^ 5233^ contributes the two rimes 7 and 5233, either of which we could use to start the next line. We chose 7 simly because it was the smallest rime we hadn t used, and we make similar choices throughout the chain. This algorithm above is often referred to as creating a factor chain. Each ossible chain of factors leads to a contradiction, i.e. the number must always become abundant. We thus can conclude 3 N. At this oint we might ask what haens if we disregard our assumtion that Q 5 is emty. This allows one more branch for each rime factor since we do not know whether we have Φ 3 q) N or Φ 5 q) N. For examle, instead of just dealing with 3 and 3 2, we also have chains involving 3 4. Surrisingly, the algorithm still successfully finishes in just under 000 stes. In articular if N is an R-OPN then 3 N. We can try to reeat this rocess for the rime 5, but the number of factors needed to achieve abundance increases dramatically. Even if we assume Q 3 is emty the number of cases to check is large; the chain must go to deths of about 80, 000 rime factors which is entirely unfeasible. We describe how to overcome these difficulties in the next section. On the other hand, for any rime larger than 5 we quickly get a smaller rime at least in ractice) and thus reduce to a revious case. For examle, the following rint-out shows that if 7 N then a smaller odd rime must divide N. When a smaller odd rime aears, we write S: 7^2 => 3^ 9^ S 7^4 => 280^

8 8 S. ADAM FLETCHER, PACE P. NIELSEN, AND PASCAL OCHEM 280^ => 2^ 3^ 467^ S 280^2 => 37^ 43^ 4933^ 37^ => 2^ 9^ 9^2 => 3^ 27^ S 9^4 => 5^ 9^ 5^2 => 3^ 7^ 093^ S 5^4 => 5^ ^ S 37^2 => 3^ 7^ 67^ S 37^4 => ^ 4^ 427^ ^2 => 7^ 9^ 9^2 => 3^ 27^ S 9^4 => 5^ 9^ 5^2 => 3^ 7^ 093^ S 5^4 => 5^ ^ S ^4 => 5^ 322^ S 280^4 => 5^ 9566^ ^ S Notice that we do not even need to involve abundance comutations. Continuing in this manner, one can rove that if 7 q < 0 8 is rime and q N then a smaller odd rime divides N. Full rint-outs of these results, and the results of the next section) are available on the second author s website. With more effort, the uer bound on q can be imroved slightly; but a large imrovement is not ossible without imroved factorization techniques which is, of course, a very difficult roblem!). Eventually the numbers we consider are too large to factor in a reasonable amount of time. It is interesting to note that Φ 3 Φ 5 x)) = x 2 x + )x 6 + 3x 5 + 5x 4 + 6x 3 + 7x 2 + 6x + 3), which hels factor some of the larger integers. Comutations suggest that any factor chain as above starting with a large rime quickly yields a smaller rime. It is an oen question whether one can create a chain using odd rimes) which eventually yields no new rime factors. 6. Dealing with the rime 5 We still need to deal with the case q = 5, and then we can conclude that an R-OPN N has no rime factors < 0 8. It should be noted that even if we assume that N is not divisible by a sixth ower the comutation is difficult because chains do not necessarily reeat rime factors. So we will continue to work with the assumtion that N is an R-OPN, even though our main interest lies in that secial case. Kee in mind that we do know that 3 N from our work in the revious section. The main difficulty in roducing factor chains starting with the rime 5 is that abundance is more difficult to achieve. When we started with the rime 3 it fortunately contributed more to the abundance comutations which heled in ending the chains quickly. To reach abundance when starting with 5 some cases will require hundreds of thousands of different rimes. It should also be mentioned that moving from the case P = {3} to the case P = {3, 5} increases the number of chains exonentially. The key insight is to notice that certain rimes occur quickly on any chain one considers. For examle, we can always eventually get the rime because of the following factor chain: 5^ => 3 S 5^2 => 3 3^2 => 3 S 3^4 => X 5^4 => X

9 SIEVE METHODS FOR ODD PERFECT NUMBERS 9 Note: To save room we do not rint the full factorization of Φ a ) to the right of the arrow, which in some cases is difficult or even imossible) to rovide since full factorization is time intensive. Instead, we merely rint the single rime factor we will use to continue the factor chain. So, for examle, the last line tells us that Φ 5 5) and that is all we need to know. Since we know 3 cannot divide N, hereafter we will dro all lines involving a 3. Now that we know that the rime must occur on any chain we can use it to get other rimes. A factor chain starting with the rime which is seventy-two lines long tells us that 3 must occur in any chain. The rime 3 quickly gives us the rime 735, and in turn we get 4 using the chain: 735^2 => ^4 => 4 X 735^4 => ^4 => ^4 => ^ => ^4 => 4 X ^4 => ^4 => 63 63^4 => 4 X Note that we do not use the contradiction S when we have a rime smaller than 735 excet on the droed lines involving 3) since this chain is really a subchain of a factor chain starting with 5. In general, the only rimes that we can cature in this way are all mod 5). This has to do with the fact that if q Φ 5 ) then q = 5 or q mod 5). This makes abundance comutations more difficult since we don t always have small rimes like 7 and 3). However, it is ossible to build a large list of rimes that must occur in all chains. The ultimate goal is to collect enough rimes, that each must occur in the chain, so that we reach abundance Since we can ignore lines giving us 3, and since 3 Φ 3 q) whenever q mod 3), one strategy that is highly effective is to branch along rimes which are mod 3) as much as ossible. This allows us to avoid having to start lines with q 2. For examle, once we have the rime we get the rime 9 using the following factor chain: ^ => 09 09^4 => 9 X ^4 => ^ => ^4 => ^4 => ^4 => ^4 => ^4 => ^4 => ^4 => ^4 => 9 X ^4 => big rime) ^4 => 9 X This strategy is extremely useful in the endgame when there are a lot of rimes with which we can start chains.

10 0 S. ADAM FLETCHER, PACE P. NIELSEN, AND PASCAL OCHEM The ossibility that a rime can be the secial rime has a quadratic effect on the number of lines in a given factor chain. But the secial rime is secial recisely because there is only one rime factor of N with its roerties. We can exloit this fact by creating two searate factor chains which involve distinct sets of ossible secial rimes. If the secial rime occurs in the first chain then it cannot aear in the second chain and vice versa. Thus, we can write both chains without any contributions from a secial rime, and at least one of them gives us the rime we need since the secial rime occurs in at most one of the two chains). To give an exlicit examle, suose we want to get the rime 05 and we already have the rimes 24 and Consider the two chains: 24^4 => ^4 => ^4 => ^4 => 05 X and ^4 => ^4 => ^4 => ^4 => ^4 => ^4 => ^4 => 05 X The only ossible secial rimes which can aear in the first chain are 24, 882 or 0642, since they are the only rimes mod 4). Thus, if the secial rime is not one of these three rimes then the first chain shows us that we obtain 05. On the other hand, if one of those three rimes is the secial rime, then the first chain is insufficient but the second chain suffices since it does not contain any of those ossible secial rimes. This rocess reduces the quadratic effect of the secial rime to a doubling effect. There is one final simlifying technique that we ut into use. Although it takes many rimes to reach abundance, if a chain does ever involve the rime 7 or other combinations of small rimes) it is much easier to reach abundance. Thus, once we know that all chains eventually contain each of the first hundred rimes congruent to mod 5), we also know that if 7 occurs in a chain we have abundance. Hence, after that oint we may ignore any line containing 7. Similarly, with enough rimes we can start ignoring lines with 3, 7, and so forth. Even before we know we can ignore 7, we know that we can ignore chains which give us both 7 and 3, or both 7 and 9. We ut this into use when trying to obtain the rime 253. We found two chains without dulicated ossible secial rimes) that either give us 253 or 7. We then found two new chains again without dulicated ossible secial rimes) that either give us 253, 3, or 9. Since 7 and 3 cannot haen together, and neither can 7 and 9, we see that we must get 253. After six months of running rograms we wrote for Mathematica, we gathered enough chains and rimes to achieve abundance. This roves that R-OPNs are not divisible by Minimizing the uer bound Let N be an R-OPN, and let C be the uer bound we constructed for the smallest rime factor of N. Recall that our first assumtion on C was made to revent N from having rime factors in common with n, d, or P. Due to the calculations above, we lose nothing in assuming C > 0 8, and we achieve the same goal. It turns out this simle fact will greatly reduce a number of constants

11 SIEVE METHODS FOR ODD PERFECT NUMBERS we deal with. For examle, in the case we are considering we have C 0 = 2. Merely assuming that C max{p max, n, C 0 } + = 6 we have 6 and so C 0 5/3. But as we have roven that C 0 8 we also have C / = which is a much better indeed, nearly otimal) constant. In all comutations hereafter we will imlicitly use the fact that N has no rime factors less than 0 8, and we will take C 0 = ). 8. Working with Q Defining Ω and κ. Our next job is to estimate q q Q 3 q. As before, we will use the large sieve. However, it turns out that we are able to imrove our sieving sets Ω if we take into account the comutations of the revious section. Notice that with n = 2 we have v 3 n) = v 5 n) = 0, and hence C =. Thus, the only ossible rime factor of N congruent to mod 5) is π, and we know π > 0 8. Let s recall some standard notation. For a rime, and an integer a Z with a, we let o a) denote the order of a modulo. In other words, this is the smallest ositive integer for which a oa) mod ). If r is a rime number and r ) then there are exactly r congruence classes mod which have order r. The abundance comutations from the revious section were only useful for small rimes. The reasons are two-fold. Primarily, when working with large rimes we can end a chain much sooner by finding a smaller rime which we will call the smaller rime contradiction). Secondarily, since we are assuming the rime factors of N are larger than 0 8 we would need an enormous number of rimes to reach abundance. Because of these two reasons we wish to encode into the definition of the sets Ω as much information as we can glean from the smaller rime contradiction, and we can ignore abundance comutations. Fix q Q 3. We then know Φ 3 q) N. The simlest ossible chain would occur if Φ 3 q) is divisible by a rime < 0 8. Such a rime is necessarily congruent to mod 3), or equal to 3, due to congruence restrictions on the ossible rime factors of Φ 3 x); see [24, Lemma ]. Thus, for all rimes < 0 8 with mod 3), we can sieve the two classes a mod ) with o a) = 3; and we can also sieve by the class mod 3). This is a finite number of congruence classes, and we will see that it only changes the bound given in the large sieve by a constant factor. The next simlest chain occurs as follows: Suose we have a rime Φ 3 q). We may as well assume q 2 mod 3) and so mod 3). We then have 3 Φ 3 ), and hence / Q 3. Therefore, either Q 5 or = π, and we can deal with the latter contingency as we did in revious sections. In the case Q 5 the chain stos as long as Φ 5 ) is divisible by a rime r < 0 8. By congruence conditions, we either have r = 5 or r mod 5). In that case either mod 5), or o r ) = 5 and belongs to one of the four congruence classes modulo r of order 5. Unlike the revious case, this gives us an infinite set of congruence classes to to sieve away. If one has good estimates for Mertens roducts over arithmetic rogressions, it would be valuable to not only sieve by rimes in the class mod 5) but in classes such as mod 3) a mod ) where a = 3, 4, 5, 9. The best exlicit bounds we are aware of, that work for moduli that get increasingly large, are found in []. Unfortunately, even in the case of reasonable sized moduli, such as k = 65, the bounds are too large to be of value in the comutations we will erform. We could continue this rocess, defining Ω to deal with chains of deeer deths. Comlications arise in this situation. For examle, we still have to deal with the ossibility that one of the factors in the chain is the secial rime. In ractice we will restrict ourselves to chains of deth one, or of deth two if they yield 3 or 5. Further note that since the the uer bound we reach for the smallest rime is we may as well relace all instances of 0 8 by the uer bound we wish to obtain. Due to technical conditions, we will work u to 0 00.

12 2 S. ADAM FLETCHER, PACE P. NIELSEN, AND PASCAL OCHEM We are now ready to define Ω. Let U be the set of rimes mod 3) which are either < 0 00 or mod 5). As just mentioned, deending on what tools are available, another choice of U may be aroriate. For an arbitrary rime, set { mod 3)} if = 3, Ω = {0 mod )} {a mod ) : o a) = 3} if U, otherwise. Define We have κ) = Ω just as before. 2 if = 3, κ) = 3 if U, otherwise Finding bounds on θx; 5, ) x/8. In the following we let θt; k, l) = t, l mod k) log). We first need to find exlicit bounds on the error term for θx; 5, ) x/8. We begin with the following result found in [27], for values of x that are small: Lemma 4. For x 0 0 we have θx; 5, ) x/ x. For slight larger x we follow [27, Section 4]. In the notation of that aer, taking R = 6.4 and C χ) = due to work in [7] which is also found in [9], and slightly imroved in the rerint [8]), we get bounds of the form θx; 5, ) x/8 < ɛx for all x > x 0, where ɛ deends on x 0. We note that, without verifying the Riemann Hyothesis to greater heights for Dirichlet L-function modulo 5, these bounds are only valid when x Here is a table of some of these values: x 0 ɛ For large values of x, we turn to the methods emloyed in [8]. In the notation of that aer, we take R = , H = 2500, and C χ) = 9.4. In [8, Theorem 5], we have X 4 = 0 and so the bounds we obtain are accurate only when logx)/r X 4, or in other words x Putting it all together, we have x logx) /4 θx; 5, ) x/8 < when x > e logx) 8.3. Bounding a Mertens tye roduct. We seek an effective bound on x, mod 5). By methods emloyed in [28], secifically in the derivation of equation 4.5) using artial summation see also [4, Lemma.3.]), we have x mod 5) = θx; 5, ) x/8 loglogx)) + M5, ) + 8 x logx) x θt; 5, ) t/8) + logt) t 2 logt) 2 dt

13 SIEVE METHODS FOR ODD PERFECT NUMBERS 3 where M5, ) = is an exlicit constant see [2]). The work in the revious section yields θx; 5, ) x/8 <.3282x/ logx) for all x 2. Plugging this into the formula above, we then have loglogx)) M5, ) 2 8 < x logx) logx). mod 5) As it will become useful shortly, we develo a series of inequalities. We have We also have 2 logx) logx) <.48 logx) when x 06. ex.48/ logx)) >.48 logx) for x 2, by a simle alication of Taylor series remainders. Similarly, for x 0 6. Next, we comute that >x, n 2 mod 5) n n < 2 >x, n 2 mod 5) ex.48/ logx)) < +.65 logx) n = 2 >x mod 5) ) < 2 n>x nn ) < 2x ) and so ex >x, n 2 mod 5) n n < ex/2x )) < +. x = αx) for x 0 6. Finally, x mod 5) αx) +.65 ) < +.66 logx) logx) when x 0 6. Taking the logarithm of a roduct, exanding the Taylor series, and taking exonentials we obtain ) = ex n n + n n. x mod 5) n 2 mod 5) Using the bounds given above, we then have, for x 0 6, C5, ).48 ) < ) < logx) /8 logx) where C5, ) = according to [22]. x mod 5) >x, n 2 mod 5) C5, ) +.66 ) logx) /8 logx)

14 4 S. ADAM FLETCHER, PACE P. NIELSEN, AND PASCAL OCHEM 8.4. Bounding another Mertens tye roduct. Again, using the methods emloyed in [28], we have = θx; 3, ) x/2 loglogx)) + M3, ) + θt; 3, ) t/2) + logt) 2 x logx) t 2 logt) 2 dt x mod 3) where M3, ) = is an exlicit constant again see [2]). By [8], we have the bound θt; 3, ) t/2 < t logt) for t 53. Reeat the rocess in the last subsection to obtain loglogx)) M3, ) 2 < x logx) logx). For x 0 6 we have mod 3) Taylor series remainders rove that and we also have Thus, for x 0 6 we have C3, ) logx) /2 0.3 logx) logx) logx) < 0.3 logx). ex 0.3/ logx)) > 0.3 logx) αx) ex0.3/ logx)) < logx) ) < where C3, ) = according to [22]. x mod 3) 8.5. Estimating B. Now, consider the quantity κ) log) logz) logz) <z <z ) < log) + log3) 3 x C3, ) ) logx) /2 logx) + <z mod 3) 2 log) A simle comutation shows that the quantity on the right is smaller than 2 for z 0 6. In the calculations below, we imlicitly assume z > 0 6 unless otherwise stated. We first estimate the iece involving the arithmetic rogression. Recall that by [8], we have the bound θt; 3, ) t/2 < t logt) for t 53. By artial summation, we comute for integer values of z that log) = θz; 3, ) z z + θt; 3, ) t 2 dt z, mod 3) which is increasing in z. In articular, logz) z, mod 3) t=7. < logz) + logz) loglogz)) log) loglogz)) logz).2 logz)

15 for all real z > 0 6. The function loglogz)) logz).2 logz) By Equation 3.22) in [28], we have SIEVE METHODS FOR ODD PERFECT NUMBERS 5 z log) < logz) + E + is bounded above by logz) for z 39, where E = is a constant. Putting all of this together, we have κ) log) < 2.02 logz) <z for all z 2. Hence, we take B = Notice that this choice of B continues to work with any of the other choices for U discussed above. Asymtotically, B = 2 would be the otimal constant when U consists of all rimes mod 3) Estimating V P w)). Another result of Dusart [7] tells us e γ 0.2 ) logw) logw) 2 < ) < e γ logw) <w Rewriting V P w)), for w 0 00 we have V P w)) = ) 2 w 0 00 mod 3) 0 00 mod 5) 0 00 mod 3) ) 3 ) logw) 2 ), for w > ) 2 ) 2 w 0 00 mod 5) ) w mod 5) 3 ) mod 5) mod 5) ) 3 ). 3 ) 2 The first two roducts in the second line are both bounded above by. The third roduct on the second line can be bounded by ) 3 ) 3 ) 3 ) = ) ) < 0 6 mod 5) mod 5) ) 3 ) <n < 0 00 mod 5) ) 3 n ) 3 < The other roducts were bounded reviously, so we have for w 0 00 that V P w)) < ) e γ 0.2 C3, ) ) + 2 logw) log0 00 ) 2 log0 00 ) ) 2 C5, )2.66 log0 00 ) /4 + logw) /4 log0 00 ) C5, ) 2 < logw) 5/4. n log0 00 ).48 log0 00 ) Recall that we must deal with the ossibility of a single element π T ; which is accomlished by multilying our answer above by ) ) 0 / 3 8 0, as π > 0 8. This quantity changes our constants 8 so little that with some forethought it can be accounted for in revious comutations. ) 2 ) 2

16 6 S. ADAM FLETCHER, PACE P. NIELSEN, AND PASCAL OCHEM 8.7. Picking a value for s > 2B. We set w = x /s, and assume w > By Proosition 2, and the work in the revious section, we have G x) e C2s)) V P x ))) /s < e C2s)) logx) 5/4 s 5/4 with C 2 s) = s 2 logs/2b) s 2 + B and B = This quantity is minimized when s is a little smaller than 0. We take s = 0 and obtain G x) 0.29 logx). 5/4 Notice that since w > 0 00 we are also assuming x > Measuring the size of Q 3. In the large sieve, taking u = x where x > 0 000, we have {q Q 3 : q < x} 2x G x) + x < 0.438x logx) 5/4. Reeating the comutation done in Section 4, for y 0 000, log q = log /q) = q q + 2q 2 + ) 3q 3 + q Q 3 q Q 3 < y q Q 3 x logx) 5/4 dx + q>y qq ) <.752 logy) + /4 nn ) <.76 logy). /4 n>y 9. Working with Q 5 We now turn our attention to the set Q 5. As many of the same comutations and remarks that were made with regards to Q 3 hold for the new set, we will simly work quickly through the aroriate bounds. We will often continue our assumtion that w > 0 00 unless otherwise osited). 9.. Defining Ω and κ. We take U to be the set of rimes q mod 5) which also either satisfy q < 0 00 or q mod 5). Set { mod 5)} if = 5, Ω = {0 mod )} {a mod ) : o a) = 5} if U, otherwise and define 2 if = 5, κ) = 5 if U, otherwise Estimating B. Comutations suggest just as before) that B = 2 is the otimal value, asymtotically. We will get close to this value. We begin with the following bounds found in [27]. Proosition 5. For t > 0 0, For all 0 < t 0 0, θt; 5, ) t 4 < t 4. θt; 5, ) t 4 < t.

17 SIEVE METHODS FOR ODD PERFECT NUMBERS 7 For integer values of z > 0 0, we then have log) = θz; 5, ) z z + z, mod 5) t= < logz) Combining this with work done in revious sections, we have κ) log) logz) logz) <z <z log) + log5) 5 for all real z > 0 0. For integer values in the range 0 5 < z < 0 0, we have z, mod 5) log) + = θz; 5, ) z + z θt; 5, ) t 2 dt <z mod 5) t= 4 log) θt; 5, ) t 2 dt < 2.05 < 0.25 logz) / z. κ) log) In articular, logz) <z 2.09 for all real z > 0 5. Finally, for z < 0 5 one checks directly that 2 works as an uer bound. We may then take B = Finding bounds on θx; 5, ) x/4. We bound θx; 5, ) x/4 using the techniques of Section 8.2. By Proosition 5, for x 0 0 we have θx; 5, ) x/ x. For slight larger x we again follow [27, Section 4]. For very large x we use [8]. Putting these comutations together we have the bound x θx; 5, ) x/8 < 0.6 logx) for all x Bounding a third Mertens tye roduct. We have the equality = θx; 5, ) x/4 loglogx)) + M5, ) + 4 x logx) x mod 5) x θt; 5, ) t/4) + logt) t 2 logt) 2 dt where M5, ) = is a constant. Using the bound given in the last subsection, we have for x 0 0 that loglogx)) M5, ) 4 < 0.9 x logx) logx). For x 0 6 we have mod 5) Letting x 0 6, Taylor remainders yield 0.9 logx) logx) < 0.67 logx). ex0.67/ logx)) < logx),

18 8 S. ADAM FLETCHER, PACE P. NIELSEN, AND PASCAL OCHEM and since by revious comutation we have we also have As we obtain x mod 5) ex ) >x, n 2 mod 5) n n < ex/2x )) < +. x αx) ) < logx) logx). = ex x mod 5) where C5, ) = according to [22]. x mod 5) ) < 9.5. Estimating V P w)). We have for w > 0 00, V P w)) < ) ) w < w mod 5) logw) 3/2. n 2 mod 5) n n + C5, ) ) logx) /4 logx) 0 00 mod 5) ) mod 5) = αx) >x, n 2 mod 5) ) 4 ) 4 n n Picking a value for s > 2B. We take s = 0 > 2B, and so x > as before. We obtain G x) 2.78 logx) 3/ Measuring the size of Q 5. In the large sieve, taking u = x where x > 0 000, we have {q Q 5 : q < x} 2x G x) + x < 5.56x logx) 3/2. Reeating the comutation done in Section 4, for y 0 000, log q = log /q) = q q + 2q 2 + ) 3q 3 + q Q 5 q Q 5 < y 5.56 q Q 5 x logx) 3/2 dx + q>y qq ) <.2 logy) + /2 nn ) <.2 logy). /2 n>y

19 SIEVE METHODS FOR ODD PERFECT NUMBERS 9 0. Putting it all together Recall that C 0 = ). Our comutations tell us that logc 0) > logy) /4 logy). /2 for y Thus y is larger than the smallest rime divisor of our urorted number, and this finishes the roof of the Main Theorem. This imroves Yamada s result by seven orders of magnitude.. Idealization One might wonder what bounds are ossible if we idealize the situation slightly. First, we may as well assume C 0 = 2. Second, we will suose that π has no further effect on the comutations. Third, let s take U to be the set of all rimes mod 3) when working with Q 3, and take U to be the set of all rimes mod 5) when working with Q 5. While Rankin s trick gives us effective bounds, it is not tight. By methods in [6, Chater ] and [0, Theorem 2.2.2], one can find an asymtotic and effective bound on Gw). In fact, for Q 3 we have Gw) c 3,g logw) 2 where c 3,g = Similarly, for Q 5 we have Gw) c 5,g logw) 2 where c 5,g = e 2γ Γ + 2) lim w logw) 2 V P w)) = e 2γ Γ + 2) lim w logw) 2 V P w)) = Fourth, let s suose we are in a sieving situation where S x/g x). Ignoring error terms we have log q q q dx q Q y x logx) 2 logy) 3 and similarly Solving the inequality q Q 3 log q Q 5 q 6.9 q logy) log2) 3 logy) yields y With a significant amount of work we could imrove the lower bound in the Main Theorem to 0 0, or ossibly to 0 2 if the comutation was distributed. This would easily surass the idealized bound we obtained above. However, that bound is still not obtainable for two reasons. First, the choices for the sets U are not realistic. Second, and more imortantly, obtaining effective asymtotic bounds for Gw) with current techniques seems to require w 0 0, and hence we are led to something close to x 0 20, which is just too large. 2. Oen questions and future directions Let be a rime and let q Φ 3 ) be a rime divisor. Either q = 3 or Φ 3 q) is divisible by 3. Similarly, if q Φ 5 ) is rime then either q = 5 or Φ 5 q) is divisible by 5. Thus, if we consider factor chains where new rimes arise only from either factoring Φ 3 ) or Φ 5 ), and we can sto whenever 3 or 5 occurs, then the factor chain simly bounces back and forth between alying Φ 3 and Φ 5. Eventually we exect

20 20 S. ADAM FLETCHER, PACE P. NIELSEN, AND PASCAL OCHEM one of the rime factors to be q mod 5), and then both Φ 3 q) and Φ 5 q) terminate our factor chain. In ractice, such chains reach rimes q mod 5) very quickly. It would be interesting to rove that for all sufficiently large rimes, factor chains limited as above always reach 3 or 5. Moreover, it would be interesting to rove or disrove) that there is a bound on the deth it takes to reach 3 or 5. One can also ask: What haens if we exand our set of limited exonents to P = {3, 5, 7}? In this case, each rime in a factor chain has three ossible branches and ossibly a fourth, if it is the secial rime). Our technique for dealing with the rime 5 is inadequate because no secific rimes aears at least quickly) in every chain, although in rincile one should still be able to deal with 3 because abundance comutations are less difficult in that case. Acknowledgements We wish to thank William Li and Carlos Pinho for their hel factoring large integers. We also thank the referee for the many suggestions which imroved our aer. References. Olivier Bordellès, An exlicit Mertens tye inequality for arithmetic rogressions, J. Inequal. Pure Al. Math ), no. 3, Article 67, 0. electronic). 2. R. P. Brent, G. L. Cohen, and H. J. J. te Riele, Imroved techniques for lower bounds for odd erfect numbers, Math. Com ), no. 96, Graeme L. Cohen and Ronald M. Sorli, On the number of distinct rime factors of an odd erfect number, J. Discrete Algorithms 2003), no., 2 35, Combinatorial algorithms. 4. Alina Carmen Cojocaru and M. Ram Murty, An introduction to sieve methods and their alications, London Mathematical Society Student Texts, vol. 66, Cambridge University Press, Cambridge, R. J. Cook, Bounds for odd erfect numbers, Number theory Ottawa, ON, 996), CRM Proc. Lecture Notes, vol. 9, Amer. Math. Soc., Providence, RI, 999, Leonard Eugene Dickson, Finiteness of the Odd Perfect and Primitive Abundant Numbers with n Distinct Prime Factors, Amer. J. Math ), no. 4, Pierre Dusart, Inégalités exlicites our ψx), θx), πx) et les nombres remiers, C. R. Math. Acad. Sci. Soc. R. Can ), no. 2, , Estimates of θx; k, l) for large values of x, Math. Com ), no. 239, electronic). 9. Takeshi Goto and Yasuo Ohno, Odd erfect numbers have a rime factor exceeding 0 8, Math. Com ), no. 263, George Greaves, Sieves in number theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 3) [Results in Mathematics and Related Areas 3)], vol. 43, Sringer-Verlag, Berlin, Otto Grün, Über ungerade vollkommene Zahlen, Math. Z ), Kevin G. Hare, New techniques for bounds on the total number of rime factors of an odd erfect number, Math. Com ), no. 260, electronic). 3. D. R. Heath-Brown, Odd erfect numbers, Math. Proc. Cambridge Philos. Soc ), no. 2, Douglas E. Iannucci, The second largest rime divisor of an odd erfect number exceeds ten thousand, Math. Com ), no. 228, , The third largest rime divisor of an odd erfect number exceeds one hundred, Math. Com ), no. 230, Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, Habiba Kadiri, An exlicit zero-free region for Dirichlet L-functions, submitted, found at htt:// kadiri/articles. 8., An exlicit zero-free region for the Dirichlet L-functions, found at htt:// 9., Une région exlicite sans zéro our les fonctions L de Dirichlet, Thèse, Université de Lille I, U.F.R. de Mathématiques - Laboratoire A.G.A.T.-U.M.R Masao Kishore, On odd erfect, quasierfect, and odd almost erfect numbers, Math. Com ), no. 54, A. Languasco and A. Zaccagnini, Comuting the Mertens and Meissel-Mertens constants for sums over arithmetic rogressions, rerint 2009).

21 SIEVE METHODS FOR ODD PERFECT NUMBERS 2 22., On the constant in the Mertens roduct for arithmetic rogressions. II. Numerical values, Math. Com ), no. 265, Wayne L. McDaniel, The non-existence of odd erfect numbers of a certain form, Arch. Math. Basel) 2 970), Pace P. Nielsen, Odd erfect numbers have at least nine distinct rime factors, Math. Com ), no. 260, electronic). 25. Paul Pollack, On Dickson s theorem concerning odd erfect number, American Math. Monthly 8 20), no. 2, Carl Pomerance, Multily erfect numbers, Mersenne rimes, and effective comutability, Math. Ann ), no. 3, Olivier Ramaré and Robert Rumely, Primes in arithmetic rogressions, Math. Com ), no. 23, J. Barkley Rosser and Lowell Schoenfeld, Aroximate formulas for some functions of rime numbers, Illinois J. Math ), Tomohiro Yamada, On the divisibility of odd erfect numbers by the ower of a rime, rerint, found at htt://arxiv.org/abs/math/0540, , Odd erfect numbers of a secial form, Colloq. Math ), no. 2, Deartment of Mathematics, Brigham Young University, Provo, UT address: adam3.459@gmail.com Deartment of Mathematics, Brigham Young University, Provo, UT address: ace@math.byu.edu CNRS, Lab. J.V. Poncelet, Moscow LRI, Bât. 490, Univ. Paris-Sud, 9405, Orsay Cedex, France address: ochem@lri.fr