Problem 1 Let Be Any Prime Number. Then, by Definition
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Paul Erdős and the Rise of Statistical Thinking in Elementary Number Theory
Paul Erd®s and the rise of statistical thinking in elementary number theory Carl Pomerance, Dartmouth College based on the joint survey with Paul Pollack, University of Georgia 1 Let us begin at the beginning: 2 Pythagoras 3 Sum of proper divisors Let s(n) be the sum of the proper divisors of n: For example: s(10) = 1 + 2 + 5 = 8; s(11) = 1; s(12) = 1 + 2 + 3 + 4 + 6 = 16: 4 In modern notation: s(n) = σ(n) − n, where σ(n) is the sum of all of n's natural divisors. The function s(n) was considered by Pythagoras, about 2500 years ago. 5 Pythagoras: noticed that s(6) = 1 + 2 + 3 = 6 (If s(n) = n, we say n is perfect.) And he noticed that s(220) = 284; s(284) = 220: 6 If s(n) = m, s(m) = n, and m 6= n, we say n; m are an amicable pair and that they are amicable numbers. So 220 and 284 are amicable numbers. 7 In 1976, Enrico Bombieri wrote: 8 There are very many old problems in arithmetic whose interest is practically nil, e.g., the existence of odd perfect numbers, problems about the iteration of numerical functions, the existence of innitely many Fermat primes 22n + 1, etc. 9 Sir Fred Hoyle wrote in 1962 that there were two dicult astronomical problems faced by the ancients. One was a good problem, the other was not so good. 10 The good problem: Why do the planets wander through the constellations in the night sky? The not-so-good problem: Why is it that the sun and the moon are the same apparent size? 11 Perfect numbers, amicable numbers, and similar topics were important to the development of elementary number theory. -
Cullen Numbers with the Lehmer Property
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0 CULLEN NUMBERS WITH THE LEHMER PROPERTY JOSE´ MAR´IA GRAU RIBAS AND FLORIAN LUCA Abstract. Here, we show that there is no positive integer n such that n the nth Cullen number Cn = n2 + 1 has the property that it is com- posite but φ(Cn) | Cn − 1. 1. Introduction n A Cullen number is a number of the form Cn = n2 + 1 for some n ≥ 1. They attracted attention of researchers since it seems that it is hard to find primes of this form. Indeed, Hooley [8] showed that for most n the number Cn is composite. For more about testing Cn for primality, see [3] and [6]. For an integer a > 1, a pseudoprime to base a is a compositive positive integer m such that am ≡ a (mod m). Pseudoprime Cullen numbers have also been studied. For example, in [12] it is shown that for most n, Cn is not a base a-pseudoprime. Some computer searchers up to several millions did not turn up any pseudo-prime Cn to any base. Thus, it would seem that Cullen numbers which are pseudoprimes are very scarce. A Carmichael number is a positive integer m which is a base a pseudoprime for any a. A composite integer m is called a Lehmer number if φ(m) | m − 1, where φ(m) is the Euler function of m. Lehmer numbers are Carmichael numbers; hence, pseudoprimes in every base. No Lehmer number is known, although it is known that there are no Lehmer numbers in certain sequences, such as the Fibonacci sequence (see [9]), or the sequence of repunits in base g for any g ∈ [2, 1000] (see [4]). -
Sums of Divisors, Perfect Numbers and Factoring*
J. COMPUT. 1986 Society for and Applied Mathematics No. 4, November 1986 02 SUMS OF DIVISORS, PERFECT NUMBERS AND FACTORING* ERIC GARY MILLER5 AND JEFFREY Abstract. Let N be a positive integer, and let denote the sum of the divisors of N = 1+2 +3+6 = 12). We show computing is equivalent to factoring N in the following sense: there is a random polynomial time algorithm that, given N),produces the prime factorization of N, and N) can be computed in polynomial time given the factorization of N. We show that the same result holds for the sum of the kth powers of divisors of We give three new examples of problems that are in Gill’s complexity class BPP perfect numbers, multiply perfect numbers, and amicable pairs. These are the first “natural” sets in BPP that are not obviously in RP. Key words. factoring, sum of divisors, perfect numbers, random reduction, multiply perfect numbers, amicable pairs subject classifications. 1. Introduction. Integer factoring is a well-known difficult problem whose precise computational complexity is still unknown. Several investigators have found algorithms that are much better than the classical method of trial division (see [Guy [Pol], [Dix], [Len]). We are interested in the relationship of factoring to other functions in number theory. It is trivial to show that classical functions like (the number of positive integers less than N and relatively prime to N) can be computed in polynomial time if one can factor N; hence computing is “easier” than factoring. One would also like to find functions “harder” than factoring. -
POWERFUL AMICABLE NUMBERS 1. Introduction Let S(N) := ∑ D Be the Sum of the Proper Divisors of the Natural Number N. Two Disti
POWERFUL AMICABLE NUMBERS PAUL POLLACK P Abstract. Let s(n) := djn; d<n d denote the sum of the proper di- visors of the natural number n. Two distinct positive integers n and m are said to form an amicable pair if s(n) = m and s(m) = n; in this case, both n and m are called amicable numbers. The first example of an amicable pair, known already to the ancients, is f220; 284g. We do not know if there are infinitely many amicable pairs. In the opposite direction, Erd}osshowed in 1955 that the set of amicable numbers has asymptotic density zero. Let ` ≥ 1. A natural number n is said to be `-full (or `-powerful) if p` divides n whenever the prime p divides n. As shown by Erd}osand 1=` Szekeres in 1935, the number of `-full n ≤ x is asymptotically c`x , as x ! 1. Here c` is a positive constant depending on `. We show that for each fixed `, the set of amicable `-full numbers has relative density zero within the set of `-full numbers. 1. Introduction P Let s(n) := djn; d<n d be the sum of the proper divisors of the natural number n. Two distinct natural numbers n and m are said to form an ami- cable pair if s(n) = m and s(m) = n; in this case, both n and m are called amicable numbers. The first amicable pair, 220 and 284, was known already to the Pythagorean school. Despite their long history, we still know very little about such pairs. -
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Arxiv:2106.08994V2 [Math.GM] 1 Aug 2021 Efc Ubr N30b.H Rvdta F2 If That and Proved Properties He Studied BC
Measuring Abundance with Abundancy Index Kalpok Guha∗ Presidency University, Kolkata Sourangshu Ghosh† Indian Institute of Technology Kharagpur, India Abstract A positive integer n is called perfect if σ(n) = 2n, where σ(n) denote n σ(n) the sum of divisors of . In this paper we study the ratio n . We de- I → I n σ(n) fine the function Abundancy Index : N Q with ( ) = n . Then we study different properties of Abundancy Index and discuss the set of Abundancy Index. Using this function we define a new class of num- bers known as superabundant numbers. Finally we study superabundant numbers and their connection with Riemann Hypothesis. 1 Introduction Definition 1.1. A positive integer n is called perfect if σ(n)=2n, where σ(n) denote the sum of divisors of n. The first few perfect numbers are 6, 28, 496, 8128, ... (OEIS A000396), This is a well studied topic in number theory. Euclid studied properties and nature of perfect numbers in 300 BC. He proved that if 2p −1 is a prime, then 2p−1(2p −1) is an even perfect number(Elements, Prop. IX.36). Later mathematicians have arXiv:2106.08994v2 [math.GM] 1 Aug 2021 spent years to study the properties of perfect numbers. But still many questions about perfect numbers remain unsolved. Two famous conjectures related to perfect numbers are 1. There exist infinitely many perfect numbers. Euler [1] proved that a num- ber is an even perfect numbers iff it can be written as 2p−1(2p − 1) and 2p − 1 is also a prime number. -
Problems Archives
Cache update: 56 minutes Problems Archives The problems archives table shows problems 1 to 651. If you would like to tackle the 10 most recently published problems then go to Recent problems. Click the description/title of the problem to view details and submit your answer. ID Description / Title Solved By 1 Multiples of 3 and 5 834047 2 Even Fibonacci numbers 666765 3 Largest prime factor 476263 4 Largest palindrome product 422115 5 Smallest multiple 428955 6 Sum square difference 431629 7 10001st prime 368958 8 Largest product in a series 309633 9 Special Pythagorean triplet 313806 10 Summation of primes 287277 11 Largest product in a grid 207029 12 Highly divisible triangular number 194069 13 Large sum 199504 14 Longest Collatz sequence 199344 15 Lattice paths 164576 16 Power digit sum 201444 17 Number letter counts 133434 18 Maximum path sum I 127951 19 Counting Sundays 119066 20 Factorial digit sum 175533 21 Amicable numbers 128681 22 Names scores 118542 23 Non-abundant sums 91300 24 Lexicographic permutations 101261 25 1000-digit Fibonacci number 137312 26 Reciprocal cycles 73631 27 Quadratic primes 76722 28 Number spiral diagonals 96208 29 Distinct powers 92388 30 Digit fifth powers 96765 31 Coin sums 74310 32 Pandigital products 62296 33 Digit cancelling fractions 62955 34 Digit factorials 82985 35 Circular primes 74645 36 Double-base palindromes 78643 37 Truncatable primes 64627 38 Pandigital multiples 55119 39 Integer right triangles 64132 40 Champernowne's constant 70528 41 Pandigital prime 59723 42 Coded triangle numbers 65704 43 Sub-string divisibility 52160 44 Pentagon numbers 50757 45 Triangular, pentagonal, and hexagonal 62652 46 Goldbach's other conjecture 53607 47 Distinct primes factors 50539 48 Self powers 100136 49 Prime permutations 50577 50 Consecutive prime sum 54478 Cache update: 56 minutes Problems Archives The problems archives table shows problems 1 to 651. -
Amicable Numbers
Amicable numbers Saud Dhaafi Mathematics Department, Western Oregon University May 11, 2021 1 / 36 • 1 is the first deficient number. • 6 is the first perfect number. • 12 is the first abundant number. • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • σ(n) is the sum of all positive divisors of a positive integer n including 1 and n itself. • A proper divisor is the function s(n) = σ(n) − n. Introduction • We know that all numbers are interesting. Zero is no amount while it is an even number. 2 / 36 • 6 is the first perfect number. • 12 is the first abundant number. • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • σ(n) is the sum of all positive divisors of a positive integer n including 1 and n itself. • A proper divisor is the function s(n) = σ(n) − n. Introduction • We know that all numbers are interesting. Zero is no amount while it is an even number. • 1 is the first deficient number. 3 / 36 • 12 is the first abundant number. • An amicable numbers are two different numbers m and n where σ(m)=σ(n) = m + n and m < n. • σ(n) is the sum of all positive divisors of a positive integer n including 1 and n itself. • A proper divisor is the function s(n) = σ(n) − n. Introduction • We know that all numbers are interesting. Zero is no amount while it is an even number. -
Some Problems of Erdős on the Sum-Of-Divisors Function
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, SERIES B Volume 3, Pages 1–26 (April 5, 2016) http://dx.doi.org/10.1090/btran/10 SOME PROBLEMS OF ERDOS˝ ON THE SUM-OF-DIVISORS FUNCTION PAUL POLLACK AND CARL POMERANCE For Richard Guy on his 99th birthday. May his sequence be unbounded. Abstract. Let σ(n) denote the sum of all of the positive divisors of n,and let s(n)=σ(n) − n denote the sum of the proper divisors of n. The functions σ(·)ands(·) were favorite subjects of investigation by the late Paul Erd˝os. Here we revisit three themes from Erd˝os’s work on these functions. First, we improve the upper and lower bounds for the counting function of numbers n with n deficient but s(n) abundant, or vice versa. Second, we describe a heuristic argument suggesting the precise asymptotic density of n not in the range of the function s(·); these are the so-called nonaliquot numbers. Finally, we prove new results on the distribution of friendly k-sets, where a friendly σ(n) k-set is a collection of k distinct integers which share the same value of n . 1. Introduction Let σ(n) be the sum of the natural number divisors of n,sothatσ(n)= d|n d. Let s(n) be the sum of only the proper divisors of n,sothats(n)=σ(n) − n. Interest in these functions dates back to the ancient Greeks, who classified numbers as deficient, perfect,orabundant according to whether s(n) <n, s(n)=n,or s(n) >n, respectively. -
ON RELATIVELY PRIME AMICABLE PAIRS 1. Introduction Distinct
ON RELATIVELY PRIME AMICABLE PAIRS PAUL POLLACK Abstract. An amicable pair consists of two distinct numbers N and M each of which is the sum of the proper divisors of the other. For example, 220 and 284 form such a pair, as do 9773505 and 11791935. While over ten million such pairs are known, we know of no pair where N and M are relatively prime. Artjuhov and Borho have shown (independently) that if one fixes an upper bound on the number of distinct prime factors of NM, then there are only finitely many such coprime amicable pairs. We prove the following entirely explicit (but impractical) upper bound: If N and M form a coprime amicable pair with !(NM) = K, then K2 NM < (2K)2 : 1. Introduction Distinct positive integers N and M are said to form an amicable pair if each is the sum of the proper divisors of the other; equivalently, σ(N) = σ(M) = M +N. Here σ(·) denotes the usual sum-of-divisors function. The smallest amicable pair, consisting of the numbers 220 and 284, was known already to the Pythagoreans (ca. 500 BCE). The extensive lore surrounding these numbers is entertainingly recounted in Dickson's History [Dic66, Chapter I], while a thorough account of modern developments can be found in the survey of Garcia, Pedersen, and te Riele [GPtR04]. In his 1917 dissertation [Gme17], Otto Gmelin noted that N and M share a nontrivial com- mon factor in all the 62 then-known amicable pairs, and he conjectured somewhat cautiously that this was a universal phenomenon. -
An Infinity of Unsolved Problems Concerning a Function in the Number Theory
FLORENTIN SMARANDACHE An Infinity Of Unsolved Problems Concerning A Function In The Number Theory In Florentin Smarandache: “Collected Papers”, vol. II. Chisinau (Moldova): Universitatea de Stat din Moldova, 1997. AN INFINITY OF UNSOLVED PROBLEMS CONCERNING A FUNCTION IN THE NUMBER THEORY §L Abstract. W.Sierpinski has asserted to an international conference that if mankind lasted for ever and numbered the unsolved problems, then in the long run all these unsolved problems would be solved. The purpose of our paper is that making an infinite number of unsolved problems to prove his sup~tion is not true. Moreover, the author considers the unsolved problems proposed in this paper can never be all solved! Every period of time has its unsolved problems which were not previously recom mended until recent progress. Number of new unsolved problems are exponentially in cresing in comparison with ancie.nt unsolved ones which are solved at present. Research into one unsolved problem may produce many new interesting problems. The reader is invited to exhibit his works about them. §2. Introduction We have constructed (*) a function '7. which associates to each non-null integer n the smallest positive integer m such that Tn! is multiple of n. Thus, if n has the standerd form: n = tpt' ... p~'. with all Pi distinct primes, all ai EN", and € = ±l, then 'l(n) = milJ'{1J,.;(a;)}, 1'$.1$.r and 'l(±l) = o. :-.row, we define the T]p functions: let P be a prime and a EN"; then 'lp(a) is a smallest positive integer b such tha.t b! is a. -
Paul Erdős and the Rise of Statistical Thinking In
PAUL ERDOS˝ AND THE RISE OF STATISTICAL THINKING IN ELEMENTARY NUMBER THEORY PAUL POLLACK AND CARL POMERANCE 1. Introduction It might be argued that elementary number theory began with Pythagoras who noted two- and-a-half millennia ago that 220 and 284 form an amicable pair. That is, if s(n) denotes the sum of the proper divisors of n (“proper divisor” means d n and 1 d < n), then | ≤ s(220) = 284 and s(284) = 220. When faced with remarkable examples such as this it is natural to wonder how special they are. Through the centuries mathematicians tried to find other examples of amicable pairs, and they did indeed succeed. But is there a formula? Are there infinitely many? In the first millennium of the common era, Thˆabit ibn Qurra came close with a formula for a subfamily of amicable pairs, but it is far from clear that his formula gives infinitely many examples and probably it does not. A special case of an amicable pair m, n is when m = n. That is, s(n)= n. These numbers are called perfect, and Euclid came up with a formula for some of them (and perhaps all of them) that probably inspired that of Thˆabit for amicable pairs. Euler showed that Euclid’s formula covers all even perfect numbers, but we still don’t know if Euclid’s formula gives infinitely many examples and our knowledge about odd perfects, even whether any exist, remains rudimentary. These are colorful and attractive problems from antiquity, but what is a modern mathe- matician to make of them? Are they just curiosities? After all, not all problems are good. -
On $ K $-Lehmer Numbers
ON k-LEHMER NUMBERS JOSE´ MAR´IA GRAU AND ANTONIO M. OLLER-MARCEN´ Abstract. Lehmer’s totient problem consists of determining the set of pos- itive integers n such that ϕ(n)|n − 1 where ϕ is Euler’s totient function. In this paper we introduce the concept of k-Lehmer number. A k-Lehmer num- ber is a composite number such that ϕ(n)|(n − 1)k. The relation between k-Lehmer numbers and Carmichael numbers leads to a new characterization of Carmichael numbers and to some conjectures related to the distribution of Carmichael numbers which are also k-Lehmer numbers. AMS 2000 Mathematics Subject Classification: 11A25,11B99 1. Introduction Lehmer’s totient problem asks about the existence of a composite number such that ϕ(n)|(n − 1), where ϕ is Euler’s totient function. Some authors denote these numbers by Lehmer numbers. In 1932, Lehmer (see [13]) showed that every Lemher numbers n must be odd and square-free, and that the number of distinct prime factors of n, d(n), must satisfy d(n) > 6. This bound was subsequently extended to d(n) > 10. The current best result, due to Cohen and Hagis (see [9]), is that n must have at least 14 prime factors and the biggest lower bound obtained for such numbers is 1030 (see [17]). It is known that there are no Lehmer numbers in certain sets, such as the Fibonacci sequence (see [15]), the sequence of repunits in base g for any g ∈ [2, 1000] (see [8]) or the Cullen numbers (see [11]).