Prime Divisors in the Rationality Condition for Odd Perfect Numbers
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1 Mersenne Primes and Perfect Numbers
1 Mersenne Primes and Perfect Numbers Basic idea: try to construct primes of the form an − 1; a, n ≥ 1. e.g., 21 − 1 = 3 but 24 − 1=3· 5 23 − 1=7 25 − 1=31 26 − 1=63=32 · 7 27 − 1 = 127 211 − 1 = 2047 = (23)(89) 213 − 1 = 8191 Lemma: xn − 1=(x − 1)(xn−1 + xn−2 + ···+ x +1) Corollary:(x − 1)|(xn − 1) So for an − 1tobeprime,weneeda =2. Moreover, if n = md, we can apply the lemma with x = ad.Then (ad − 1)|(an − 1) So we get the following Lemma If an − 1 is a prime, then a =2andn is prime. Definition:AMersenne prime is a prime of the form q =2p − 1,pprime. Question: are they infinitely many Mersenne primes? Best known: The 37th Mersenne prime q is associated to p = 3021377, and this was done in 1998. One expects that p = 6972593 will give the next Mersenne prime; this is close to being proved, but not all the details have been checked. Definition: A positive integer n is perfect iff it equals the sum of all its (positive) divisors <n. Definition: σ(n)= d|n d (divisor function) So u is perfect if n = σ(u) − n, i.e. if σ(u)=2n. Well known example: n =6=1+2+3 Properties of σ: 1. σ(1) = 1 1 2. n is a prime iff σ(n)=n +1 p σ pj p ··· pj pj+1−1 3. If is a prime, ( )=1+ + + = p−1 4. (Exercise) If (n1,n2)=1thenσ(n1)σ(n2)=σ(n1n2) “multiplicativity”. -
On Perfect and Multiply Perfect Numbers
On perfect and multiply perfect numbers . by P. ERDÖS, (in Haifa . Israel) . Summary. - Denote by P(x) the number o f integers n ~_ x satisfying o(n) -- 0 (mod n.), and by P2 (x) the number of integers nix satisfying o(n)-2n . The author proves that P(x) < x'314:4- and P2 (x) < x(t-c)P for a certain c > 0 . Denote by a(n) the sum of the divisors of n, a(n) - E d. A number n din is said to be perfect if a(n) =2n, and it is said to be multiply perfect if o(n) - kn for some integer k . Perfect numbers have been studied since antiquity. l t is contained in the books of EUCLID that every number of the form 2P- ' ( 2P - 1) where both p and 2P - 1 are primes is perfect . EULER (1) proved that every even perfect number is of the above form . It is not known if there are infinitely many even perfect numbers since it is not known if there are infinitely many primes of the form 2P - 1. Recently the electronic computer of the Institute for Numerical Analysis the S .W.A .C . determined all primes of the form 20 - 1 for p < 2300. The largest prime found was 2J9 "' - 1, which is the largest prime known at present . It is not known if there are an odd perfect numbers . EULER (') proved that all odd perfect numbers are of the form (1) pam2, p - x - 1 (mod 4), and SYLVESTER (') showed that an odd perfect number must have at least five distinct prime factors . -
Unit 6: Multiply & Divide Fractions Key Words to Know
Unit 6: Multiply & Divide Fractions Learning Targets: LT 1: Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b) LT 2: Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g. by using visual fraction models or equations to represent the problem. LT 3: Apply and extend previous understanding of multiplication to multiply a fraction or whole number by a fraction. LT 4: Interpret the product (a/b) q ÷ b. LT 5: Use a visual fraction model. Conversation Starters: Key Words § How are fractions like division problems? (for to Know example: If 9 people want to shar a 50-lb *Fraction sack of rice equally by *Numerator weight, how many pounds *Denominator of rice should each *Mixed number *Improper fraction person get?) *Product § 3 pizzas at 10 slices each *Equation need to be divided by 14 *Division friends. How many pieces would each friend receive? § How can a model help us make sense of a problem? Fractions as Division Students will interpret a fraction as division of the numerator by the { denominator. } What does a fraction as division look like? How can I support this Important Steps strategy at home? - Frac&ons are another way to Practice show division. https://www.khanacademy.org/math/cc- - Fractions are equal size pieces of a fifth-grade-math/cc-5th-fractions-topic/ whole. tcc-5th-fractions-as-division/v/fractions- - The numerator becomes the as-division dividend and the denominator becomes the divisor. Quotient as a Fraction Students will solve real world problems by dividing whole numbers that have a quotient resulting in a fraction. -
Arxiv:1412.5226V1 [Math.NT] 16 Dec 2014 Hoe 11
q-PSEUDOPRIMALITY: A NATURAL GENERALIZATION OF STRONG PSEUDOPRIMALITY JOHN H. CASTILLO, GILBERTO GARC´IA-PULGAR´IN, AND JUAN MIGUEL VELASQUEZ-SOTO´ Abstract. In this work we present a natural generalization of strong pseudoprime to base b, which we have called q-pseudoprime to base b. It allows us to present another way to define a Midy’s number to base b (overpseudoprime to base b). Besides, we count the bases b such that N is a q-probable prime base b and those ones such that N is a Midy’s number to base b. Furthemore, we prove that there is not a concept analogous to Carmichael numbers to q-probable prime to base b as with the concept of strong pseudoprimes to base b. 1. Introduction Recently, Grau et al. [7] gave a generalization of Pocklignton’s Theorem (also known as Proth’s Theorem) and Miller-Rabin primality test, it takes as reference some works of Berrizbeitia, [1, 2], where it is presented an extension to the concept of strong pseudoprime, called ω-primes. As Grau et al. said it is right, but its application is not too good because it is needed m-th primitive roots of unity, see [7, 12]. In [7], it is defined when an integer N is a p-strong probable prime base a, for p a prime divisor of N −1 and gcd(a, N) = 1. In a reading of that paper, we discovered that if a number N is a p-strong probable prime to base 2 for each p prime divisor of N − 1, it is actually a Midy’s number or a overpseu- doprime number to base 2. -
MORE-ON-SEMIPRIMES.Pdf
MORE ON FACTORING SEMI-PRIMES In the last few years I have spent some time examining prime numbers and their properties. Among some of my new results are the a Prime Number Function F(N) and the concept of Number Fraction f(N). We can define these quantities as – (N ) N 1 f (N 2 ) 1 f (N ) and F(N ) N Nf (N 3 ) Here (N) is the divisor function of number theory. The interesting property of these functions is that when N is a prime then f(N)=0 and F(N)=1. For composite numbers f(N) is a positive fraction and F(N) will be less than one. One of the problems of major practical interest in number theory is how to rapidly factor large semi-primes N=pq, where p and q are prime numbers. This interest stems from the fact that encoded messages using public keys are vulnerable to decoding by adversaries if they can factor large semi-primes when they have a digit length of the order of 100. We want here to show how one might attempt to factor such large primes by a brute force approach using the above f(N) function. Our starting point is to consider a large semi-prime given by- N=pq with p<sqrt(N)<q By the basic definition of f(N) we get- p q p2 N f (N ) f ( pq) N pN This may be written as a quadratic in p which reads- p2 pNf (N ) N 0 It has the solution- p K K 2 N , with p N Here K=Nf(N)/2={(N)-N-1}/2. -
And Its Properties on the Sequence Related to Lucas Numbers
Mathematica Aeterna, Vol. 2, 2012, no. 1, 63 - 75 On the sequence related to Lucas numbers and its properties Cennet Bolat Department of Mathematics, Art and Science Faculty, Mustafa Kemal University, 31034, Hatay,Turkey Ahmet I˙pek Department of Mathematics, Art and Science Faculty, Mustafa Kemal University, 31034, Hatay,Turkey Hasan Köse Department of Mathematics, Science Faculty, Selcuk University 42031, Konya,Turkey Abstract The Fibonacci sequence has been generalized in many ways, some by preserving the initial conditions, and others by preserving the recur- rence relation. In this article, we study a new generalization {Lk,n}, with initial conditions Lk,0 = 2 and Lk,1 = 1, which is generated by the recurrence relation Lk,n = kLk,n−1 + Lk,n−2 for n ≥ 2, where k is integer number. Some well-known sequence are special case of this generalization. The Lucas sequence is a special case of {Lk,n} with k = 1. Modified Pell-Lucas sequence is {Lk,n} with k = 2. We produce an extended Binet’s formula for {Lk,n} and, thereby, identities such as Cassini’s, Catalan’s, d’Ocagne’s, etc. using matrix algebra. Moreover, we present sum formulas concerning this new generalization. Mathematics Subject Classi…cation: 11B39, 15A23. Keywords: Classic Fibonacci numbers; Classic Lucas numbers; k-Fibonacci numbers; k-Lucas numbers, Matrix algebra. 64 C. Bolat, A. Ipek and H. Kose 1 Introduction In recent years, many interesting properties of classic Fibonacci numbers, clas- sic Lucas numbers and their generalizations have been shown by researchers and applied to almost every …eld of science and art. -
Appendix a Tables of Fermat Numbers and Their Prime Factors
Appendix A Tables of Fermat Numbers and Their Prime Factors The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. Carl Friedrich Gauss Disquisitiones arithmeticae, Sec. 329 Fermat Numbers Fo =3, FI =5, F2 =17, F3 =257, F4 =65537, F5 =4294967297, F6 =18446744073709551617, F7 =340282366920938463463374607431768211457, Fs =115792089237316195423570985008687907853 269984665640564039457584007913129639937, Fg =134078079299425970995740249982058461274 793658205923933777235614437217640300735 469768018742981669034276900318581864860 50853753882811946569946433649006084097, FlO =179769313486231590772930519078902473361 797697894230657273430081157732675805500 963132708477322407536021120113879871393 357658789768814416622492847430639474124 377767893424865485276302219601246094119 453082952085005768838150682342462881473 913110540827237163350510684586298239947 245938479716304835356329624224137217. The only known Fermat primes are Fo, ... , F4 • 208 17 lectures on Fermat numbers Completely Factored Composite Fermat Numbers m prime factor year discoverer 5 641 1732 Euler 5 6700417 1732 Euler 6 274177 1855 Clausen 6 67280421310721* 1855 Clausen 7 59649589127497217 1970 Morrison, Brillhart 7 5704689200685129054721 1970 Morrison, Brillhart 8 1238926361552897 1980 Brent, Pollard 8 p**62 1980 Brent, Pollard 9 2424833 1903 Western 9 P49 1990 Lenstra, Lenstra, Jr., Manasse, Pollard 9 p***99 1990 Lenstra, Lenstra, Jr., Manasse, Pollard -
ON PERFECT and NEAR-PERFECT NUMBERS 1. Introduction a Perfect
ON PERFECT AND NEAR-PERFECT NUMBERS PAUL POLLACK AND VLADIMIR SHEVELEV Abstract. We call n a near-perfect number if n is the sum of all of its proper divisors, except for one of them, which we term the redundant divisor. For example, the representation 12 = 1 + 2 + 3 + 6 shows that 12 is near-perfect with redundant divisor 4. Near-perfect numbers are thus a very special class of pseudoperfect numbers, as defined by Sierpi´nski. We discuss some rules for generating near-perfect numbers similar to Euclid's rule for constructing even perfect numbers, and we obtain an upper bound of x5=6+o(1) for the number of near-perfect numbers in [1; x], as x ! 1. 1. Introduction A perfect number is a positive integer equal to the sum of its proper positive divisors. Let σ(n) denote the sum of all of the positive divisors of n. Then n is a perfect number if and only if σ(n) − n = n, that is, σ(n) = 2n. The first four perfect numbers { 6, 28, 496, and 8128 { were known to Euclid, who also succeeded in establishing the following general rule: Theorem A (Euclid). If p is a prime number for which 2p − 1 is also prime, then n = 2p−1(2p − 1) is a perfect number. It is interesting that 2000 years passed before the next important result in the theory of perfect numbers. In 1747, Euler showed that every even perfect number arises from an application of Euclid's rule: Theorem B (Euler). All even perfect numbers have the form 2p−1(2p − 1), where p and 2p − 1 are primes. -
NOTES on CARTIER and WEIL DIVISORS Recall: Definition 0.1. A
NOTES ON CARTIER AND WEIL DIVISORS AKHIL MATHEW Abstract. These are notes on divisors from Ravi Vakil's book [2] on scheme theory that I prepared for the Foundations of Algebraic Geometry seminar at Harvard. Most of it is a rewrite of chapter 15 in Vakil's book, and the originality of these notes lies in the mistakes. I learned some of this from [1] though. Recall: Definition 0.1. A line bundle on a ringed space X (e.g. a scheme) is a locally free sheaf of rank one. The group of isomorphism classes of line bundles is called the Picard group and is denoted Pic(X). Here is a standard source of line bundles. 1. The twisting sheaf 1.1. Twisting in general. Let R be a graded ring, R = R0 ⊕ R1 ⊕ ::: . We have discussed the construction of the scheme ProjR. Let us now briefly explain the following additional construction (which will be covered in more detail tomorrow). L Let M = Mn be a graded R-module. Definition 1.1. We define the sheaf Mf on ProjR as follows. On the basic open set D(f) = SpecR(f) ⊂ ProjR, we consider the sheaf associated to the R(f)-module M(f). It can be checked easily that these sheaves glue on D(f) \ D(g) = D(fg) and become a quasi-coherent sheaf Mf on ProjR. Clearly, the association M ! Mf is a functor from graded R-modules to quasi- coherent sheaves on ProjR. (For R reasonable, it is in fact essentially an equiva- lence, though we shall not need this.) We now set a bit of notation. -
+P.+1 P? '" PT Hence
1907.] MULTIPLY PEKFECT NUMBERS. 383 A TABLE OF MULTIPLY PERFECT NUMBERS. BY PEOFESSOR R. D. CARMICHAEL. (Read before the American Mathematical Society, February 23, 1907.) A MULTIPLY perfect number is one which is an exact divisor of the sum of all its divisors, the quotient being the multiplicity.* The object of this paper is to exhibit a method for determining all such numbers up to 1,000,000,000 and to give a complete table of them. I include an additional table giving such other numbers as are known to me to be multiply perfect. Let the number N, of multiplicity m (m> 1), be of the form where pv p2, • • •, pn are different primes. Then by definition and by the formula for the sum of the factors of a number, we have +P? -1+---+p + ~L pi" + Pi»'1 + • • (l)m=^- 1 •+P.+1 p? '" PT Hence (2) Pi~l Pa"1 Pn-1 These formulas will be of frequent use throughout the paper. Since 2•3•5•7•11•13•17•19•23 = 223,092,870, multiply perfect numbers less than 1,000,000,000 contain not more than nine different prime factors ; such numbers, lacking the factor 2, contain not more than eight different primes ; and, lacking the factors 2 and 3, they contain not more than seven different primes. First consider the case in which N does not contain either 2 or 3 as a factor. By equation (2) we have (^\ «M ^ 6 . 7 . 11 . 1£. 11 . 19 .2ft « 676089 [Ö) m<ï*6 TTT if 16 ï¥ ïî — -BTÏTT6- Hence m=2 ; moreover seven primes are necessary to this value. -
On Arithmetic Functions of Balancing and Lucas-Balancing Numbers 1
MATHEMATICAL COMMUNICATIONS 77 Math. Commun. 24(2019), 77–81 On arithmetic functions of balancing and Lucas-balancing numbers Utkal Keshari Dutta and Prasanta Kumar Ray∗ Department of Mathematics, Sambalpur University, Jyoti Vihar, Burla 768 019, Sambalpur, Odisha, India Received January 12, 2018; accepted June 21, 2018 Abstract. For any integers n ≥ 1 and k ≥ 0, let φ(n) and σk(n) denote the Euler phi function and the sum of the k-th powers of the divisors of n, respectively. In this article, the solutions to some Diophantine equations about these functions of balancing and Lucas- balancing numbers are discussed. AMS subject classifications: 11B37, 11B39, 11B83 Key words: Euler phi function, divisor function, balancing numbers, Lucas-balancing numbers 1. Introduction A balancing number n and a balancer r are the solutions to a simple Diophantine equation 1+2+ + (n 1)=(n +1)+(n +2)+ + (n + r) [1]. The sequence ··· − ··· of balancing numbers Bn satisfies the recurrence relation Bn = 6Bn Bn− , { } +1 − 1 n 1 with initials (B ,B ) = (0, 1). The companion of Bn is the sequence of ≥ 0 1 { } Lucas-balancing numbers Cn that satisfies the same recurrence relation as that { } of balancing numbers but with different initials (C0, C1) = (1, 3) [4]. Further, the closed forms known as Binet formulas for both of these sequences are given by n n n n λ1 λ2 λ1 + λ2 Bn = − , Cn = , λ λ 2 1 − 2 2 where λ1 =3+2√2 and λ2 =3 2√2 are the roots of the equation x 6x +1=0. Some more recent developments− of balancing and Lucas-balancing numbers− can be seen in [2, 3, 6, 8]. -
Arxiv:1606.08690V5 [Math.NT] 27 Apr 2021 on Prime Factors of Mersenne
On prime factors of Mersenne numbers Ady Cambraia Jr,∗ Michael P. Knapp,† Ab´ılio Lemos∗, B. K. Moriya∗ and Paulo H. A. Rodrigues‡ [email protected] [email protected] [email protected] [email protected] paulo [email protected] April 29, 2021 Abstract n Let (Mn)n≥0 be the Mersenne sequence defined by Mn = 2 − 1. Let ω(n) be the number of distinct prime divisors of n. In this short note, we present a description of the Mersenne numbers satisfying ω(Mn) ≤ 3. Moreover, we prove that the inequality, (1−ǫ) log log n given ǫ> 0, ω(Mn) > 2 − 3 holds for almost all positive integers n. Besides, a we present the integer solutions (m, n, a) of the equation Mm+Mn = 2p with m,n ≥ 2, p an odd prime number and a a positive integer. 2010 Mathematics Subject Classification: 11A99, 11K65, 11A41. Keywords: Mersenne numbers, arithmetic functions, prime divisors. 1 Introduction arXiv:1606.08690v5 [math.NT] 27 Apr 2021 n Let (Mn)n≥0 be the Mersenne sequence defined by Mn = 2 − 1, for n ≥ 0. A simple argument shows that if Mn is a prime number, then n is a prime number. When Mn is a prime number, it is called a Mersenne prime. Throughout history, many researchers sought to find Mersenne primes. Some tools are very important for the search for Mersenne primes, mainly the Lucas-Lehmer test. There are papers (see for example [1, 5, 21]) that seek to describe the prime factors of Mn, where Mn is a composite number and n is a prime number.