DOCSLIB.ORG

About Connection of One Matrix of Composite Numbers with Legendre's Conjecture

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
About Connection of One Matrix of Composite Numbers with Legendre's Conjecture About connection of one matrix of composite numbers with Legendre’s conjecture Garipov Ilshat Ilsurovich Russia, Republic of Tatarstan, Naberezhnye Chelny e-mail: [email protected]. Abstract Scientific paper is devoted to establish connection of T -matrix – matrix of composite numbers 6h ± 1 in special view – with Legendre’s conjecture. Keywords: T -matrix, prime numbers, leading element, upper and lower defining elements of number, ¾active¿ set for numbers (m − 1)4 and m4, ¾critical¿ element for numbers (m − 1)4 and m4, Legendre’s conjecture, ¾Weak¿ and ¾Strong¿ conjectures. List of symbols N0 – set of all natural numbers with zero. N – set of all natural numbers. Z – set of all integers. P – set of all prime numbers. R – set of all real numbers. T – matrix comprising all defining and all not defining elements. Te – set of all elements of T-matrix. D(b) – T -matrix upper defining element of number b. d(b) – T -matrix lower defining element of number b. Dk(b) – upper defining element of number b in k-row (k > 1) of T -matrix. dk(b) – lower defining element of number b in k-row (k > 1) of T -matrix. W (b) – T -matrix upper element of number b. arXiv:2104.06261v1 [math.GM] 7 Apr 2021 w(b) – T -matrix lower element of number b. Wk(b) – upper element of number b in k-row (k > 1) of T -matrix. wk(b) – lower element of number b in k-row (k > 1) of T -matrix. DT – set of all defining elements of T -matrix. nDT – set of all not defining elements of T -matrix. MT – set of all leading elements of T -matrix. DTk – set of all defining elements in k-row (k > 1) of T -matrix. π(x) – function counting the number of prime numbers less than or equal to x 2 R. 1 πMT (x) – function counting the number of T -matrix leading elements less than or equal to x 2 R. #k(a) – number of element a in k-row of T -matrix. 4 4 H(m−1)4; m4 – ¾active¿ set for numbers (m − 1) ; m . 4 4 C(m−1)4; m4 – ¾critical¿ element for numbers (m − 1) ; m . νk(x) – function counting the number of elements, less than or equal to x 2 R, in k-row of T -matrix. ν(x) – function counting the number of naturals of the form 6h ± 1, less than or equal to x 2 R. a%b – remainder after dividing a 2 N by b 2 N. 2 2 qm – number of prime numbers between m and (m + 1) . Introduction 1. T -matrix We construct a matrix T ≡ (a(k; n))∞×∞, where a(k; n) is a T -matrix element located in k-th row, n-th column and defined as follows: jnk n − 1 a(k; n) ≡ p(k) · 5 + 2 · + 4 · ; 2 2 1 where p(k) is the k-th element of sequence (p(k))k=1 of prime numbers: p(k) ≡ pk+2; (1) where pi is the i-th prime number in sequence of all prime numbers (see [1]). 1 Let (f(n))n=1 is a numerical sequence, where a common member f(n) is defined as follows: 3 − (−1)n f(n) ≡ 3n + : 2 THEOREM 1.1. (8k;n 2 N)(a(k; n) = p(k) · f(n)) : (2) DEFINITION 1.1. An element a(k; n) of matrix T is called defining if 1) a(k; n) is not divisible by 5; 2) a(k; n) can be expressed as a product of some two prime numbers, that is 5 6 ja(k; n) ^ (9p1;p2 2 P)(a(k; n) = p1 · p2): (3) DEFINITION 1.2. An element a(k; n) of matrix T is called not defining if he does not satisfy condition (3). DEFINITION 1.3. An element a(k; n) of matrix T is called leading if a(k; n) = p2(k): 2 DEFINITION 1.4. A T -matrix is called matrix comprising all defining and not defining elements. 1 LEMMA 1.2. (f(n))n=1 is a sequence of all numbers of the form 6h ± 1: 5; 7; 11; 13; 17; 19; 23; 25; ::: ; 6h − 1; 6h + 1; :::: 2 1 PROPERTY 1.1. The sequence (p (k))k=1 of T -matrix leading elements is ascending. The simplest properties and basic theorems about elements of T -matrix are proved in [1]. 2. About a T -matrix upper defining element of real number DEFINITION 2.1. A T -matrix defining element D(b) is called an upper defining element of number b 2 R : b > 49, if D(b) = min a(k1; n); a(k1;n)2DT a(k1;n)>b n2N where k1 is defined by condition 2 2 p (k1) = max p (k): 2 p (k)6b k>1 DEFINITION 2.2. A T -matrix defining element d(b) is called a lower defining element of number b 2 R : b > 49, if d(b) = max a(k2; n); a(k2;n)2DT a(k2;n)<b n2N where k2 is defined by condition 2 2 p (k2) = max p (k): p2(k)<b k>1 DEFINITION 2.3. A T -matrix defining element Dk(b) is called an upper defining element 2 of number b 2 R : p (k) 6 b, in k-row (k > 1) of T -matrix if Dk(b) = min a(k; n): a(k;n)2DT a(k;n)>b n2N DEFINITION 2.4. A T -matrix defining element dk(b) is called a lower defining element of number b 2 R : p2(k) < b, in k-row (k > 1) of T -matrix if dk(b) = max a(k; n): a(k;n)2DT a(k;n)<b n2N DEFINITION 2.5. A T -matrix element W (b) is called an upper element of number b 2 R : b > 49, if 3 W (b) = min a(k1; n); a(k1;n)2Te a(k1;n)>b n2N where k1 is defined by condition 2 2 p (k1) = max p (k): 2 p (k)6b k>1 DEFINITION 2.6. A T -matrix element w(b) is called a lower element of number b 2 R : b > 49, if w(b) = max a(k2; n); a(k2;n)2Te a(k2;n)<b n2N where k2 is defined by condition 2 2 p (k2) = max p (k): p2(k)<b k>1 DEFINITION 2.7. A T -matrix element Wk(b) is called an upper element of number b 2 R : 2 p (k) 6 b, in k-row (k > 1) of T -matrix if Wk(b) = min a(k; n): a(k;n)2Te a(k;n)>b n2N DEFINITION 2.8. A T -matrix element wk(b) is called a lower element of number b 2 R : p2(k) < b, in k-row (k > 1) of T -matrix if wk(b) = max a(k; n): a(k;n)2Te a(k;n)<b n2N LEMMA 2.1. (8k > 1) (8n > 1) (a(k; n) 2 DT , f(n) 2 Pnf2; 3; 5g) : (4) PROOF. Choose any k-row (k > 1) and any n-column (n > 1) of T -matrix. Necessity. Let a(k; n) 2 DT . By Definition 1.1, that means that 5 6 ja(k; n) ^ (9 p1; p2 2 P)(a(k; n) = p1 · p2): Then by rule (1), Lemma 1.2 and Theorem 1.1, p1 = p(k) 2 Pnf2; 3; 5g ^ p2 = f(n) 2 Pnf2; 3; 5g: It follows that f(n) 2 Pnf2; 3; 5g. The necessity is proved. 4 Sufficiency. Let f(n) 2 Pnf2; 3; 5g. It is clear that p(k) 2 Pnf2; 3; 5g for k > 1. p(k); f(n) 2 Pnf2; 3; 5g ) (2) ) p(k) · f(n) = a(k; n) ^ p(k); f(n) 2 P ^ 5 6 ja(k; n) ) (3) ) 5 6 ja(k; n) ^ (9 p1;p2 2 P)(a(k; n) = p1 · p2) , a(k; n) 2 DT : The sufficiency is proved. Lemma 2.1 is proved. THEOREM 2.2 (about the ¾transition down¿ of T -matrix defining element). 2 (8k; n 2 N) p (k) < a(k; n) ^ a(k; n) 2 DT ) 2 2 2 ) (9! j 2 N) k < j ^ a(k; n) < p (j) ^ a(j;#k(p (k))) = a(k; n) ^ a(j; n) = p (j) : (5) PROOF. Existence. It is established in [1]. Uniqueness. Suppose, 2 (8k; n 2 N)(p (k) < a(k; n) ^ a(k; n) 2 DT ) 2 2 ) (9 j1; j2 2 N)(j1 6= j2 ^ k < j1 ^ k < j2 ^ a(k; n) < p (j1) ^ a(k; n) < p (j2) ^ 2 2 ^ a(j1;#k(p (k))) = a(k; n) ^ a(j2;#k(p (k))) = a(k; n) ^ 2 2 ^ a(j1; n) = p (j1) ^ a(j2; n) = p (j2))): 2 2 a(j1;#k(p (k))) = a(k; n) ^ a(j2;#k(p (k))) = a(k; n) ^ j1 6= j2 ) 2 2 (2) ) a(j1;#k(p (k))) = a(j2;#k(p (k))) ^ j1 6= j2 , 2 2 , p(j1) · f(#k(p (k))) = p(j2) · f(#k(p (k))) ^ j1 6= j2 , , p(j1) = p(j2) ^ j1 6= j2 , j1 = j2 ^ j1 6= j2: As a result, a contradiction. The uniqueness is established. Theorem 2.2 is proved. COROLLARY 2.3. Let a(k; n), a(j; n) are T -matrix elements from Theorem 2.2. Then, a(k; n) a(k; n) − p(k) = p(j) − : p(k) p(j) PROOF. From algorithm №1 in [1], we get a(k; n) (9h 2 ) p2(k) + 2h · p(k) = a(k; n) , (9h 2 ) p(k) + 2h = : N N p(k) 5 2 2 a(k; n) (5) a(j;# (p (k))) (2) p(j) · f(# (p (k))) 2h = − p(k) = k − p(k) = k − p(k) = p(k) p(k) p(k) 2 p(j) · p(k) (2) a(j;# (p (k))) (5) a(k; n) = − f(# (p2(k))) = p(j) − k = p(j) − : p(k) k p(j) p(j) Corollary 2.3 is proved.
Load more

Recommended publications
  • A Potentially Fast Primality Test
    A Potentially Fast Primality Test Tsz-Wo Sze February 20, 2007 1 Introduction In 2002, Agrawal, Kayal and Saxena [3] gave the ¯rst deterministic, polynomial- time primality testing algorithm. The main step was the following. Theorem 1.1. (AKS) Given an integer n > 1, let r be an integer such that 2 ordr(n) > log n. Suppose p (x + a)n ´ xn + a (mod n; xr ¡ 1) for a = 1; ¢ ¢ ¢ ; b Á(r) log nc: (1.1) Then, n has a prime factor · r or n is a prime power. The running time is O(r1:5 log3 n). It can be shown by elementary means that the required r exists in O(log5 n). So the running time is O(log10:5 n). Moreover, by Fouvry's Theorem [8], such r exists in O(log3 n), so the running time becomes O(log7:5 n). In [10], Lenstra and Pomerance showed that the AKS primality test can be improved by replacing the polynomial xr ¡ 1 in equation (1.1) with a specially constructed polynomial f(x), so that the degree of f(x) is O(log2 n). The overall running time of their algorithm is O(log6 n). With an extra input integer a, Berrizbeitia [6] has provided a deterministic primality test with time complexity 2¡ min(k;b2 log log nc)O(log6 n), where 2kjjn¡1 if n ´ 1 (mod 4) and 2kjjn + 1 if n ´ 3 (mod 4). If k ¸ b2 log log nc, this algorithm runs in O(log4 n). The algorithm is also a modi¯cation of AKS by verifying the congruent equation s (1 + mx)n ´ 1 + mxn (mod n; x2 ¡ a) for a ¯xed s and some clever choices of m¡ .¢ The main drawback of this algorithm¡ ¢ is that it requires the Jacobi symbol a = ¡1 if n ´ 1 (mod 4) and a = ¡ ¢ n n 1¡a n = ¡1 if n ´ 3 (mod 4).
    [Show full text]
  • Primality Testing and Sub-Exponential Factorization
    Primality Testing and Sub-Exponential Factorization David Emerson Advisor: Howard Straubing Boston College Computer Science Senior Thesis May, 2009 Abstract This paper discusses the problems of primality testing and large number factorization. The first section is dedicated to a discussion of primality test- ing algorithms and their importance in real world applications. Over the course of the discussion the structure of the primality algorithms are devel- oped rigorously and demonstrated with examples. This section culminates in the presentation and proof of the modern deterministic polynomial-time Agrawal-Kayal-Saxena algorithm for deciding whether a given n is prime. The second section is dedicated to the process of factorization of large com- posite numbers. While primality and factorization are mathematically tied in principle they are very di⇥erent computationally. This fact is explored and current high powered factorization methods and the mathematical structures on which they are built are examined. 1 Introduction Factorization and primality testing are important concepts in mathematics. From a purely academic motivation it is an intriguing question to ask how we are to determine whether a number is prime or not. The next logical question to ask is, if the number is composite, can we calculate its factors. The two questions are invariably related. If we can factor a number into its pieces then it is obviously not prime, if we can’t then we know that it is prime. The definition of primality is very much derived from factorability. As we progress through the known and developed primality tests and factorization algorithms it will begin to become clear that while primality and factorization are intertwined they occupy two very di⇥erent levels of computational di⇧culty.
    [Show full text]
  • 1 Problem Summary & Motivation 2 Previous Work
    Primality Testing Report CS6150 December 9th, 2011 Steven Lyde, Ryan Lindeman, Phillip Mates 1 Problem Summary & Motivation Primes are used pervasively throughout modern society. E-commerce, privacy, and psuedorandom number generators all make heavy use of primes. There currently exist many different primality tests, each with their distinct advantages and disadvantages. These tests fall into two general categories, probabilistic and deterministic. Probabilistic algorithms determine with a certain probability that a number is prime, while deterministic algorithms prove whether a number is prime or not. Probabilistic algorithms are fast but contain error. Deterministic algorithms are much slower but produce proofs of correctness with no potential for error. Furthermore, some deterministic algorithms don't have proven upper bounds on their run time complexities, relying on heuristics. How can we combine these algorithms to create a fast, deterministic primality proving algorithm? Is there room for improvement of existing algorithms by hybridridizing them? This is an especially interesting question due to the fact that AKS is relatively new and unexplored. These are the questions we will try to answer in this report. 2 Previous Work For this project we focused on three algorithms, Miller-Rabin, Elliptic Curve Primality Proving (ECPP), and AKS. What is the state of the art for these different implementations? 2.1 ECPP Originally a purely theoretical construct, elliptic curves have recently proved themselves useful in many com- putational applications. Their elegance and power provide considerable leverage in primality proving and factorization studies [6]. From our reference [6], the following describes Elliptic curve fundamentals. Consider the following two or three variable homogeneous equations of degree-3: ax3 + bx2y + cxy2 + dy3 + ex2 + fxy + gy2 + hx + iy + j = 0 ax3 + bx2y + cxy2 + dy3 + ex2z + fxyz + gy2z + hxz2 + iyz2 + jz3 = 0 For these to be degree-3 polynomials, we must ensure at least a, b, c, d, or j is nonzero.
    [Show full text]
  • Comparing AKS Algorithm for Deterministic Primality Testing Using Matlab Updated with Mapple and C++ 1Chidozie, K
    Futo Journal Series (FUTOJNLS) e-ISSN : 2476-8456 p-ISSN : 2467-8325 Volume-5, Issue-1, pp- 189 - 209 www.futojnls.org Research Paper July 2019 Comparing AKS Algorithm for Deterministic Primality Testing Using Matlab Updated With Mapple and C++ 1Chidozie, K. K, 1*Obi, M. C. and 2Nwamba, I. 1Department of Mathematics, Federal University of Technology, Owerri, Imo State, Nigeria 2ICT Department, Imo State University, Owerri, Imo State, Nigeria *Corresponding Author’s Email: [email protected]; [email protected] Abstract This work compared the use of MATLAB updated with MAPPLE and C++ to implement AKS algorithm which is the first deterministic primality testing algorithm in polynomial time. To test if the number 9965468763136528274628451 is prime took about 210 minutes in AKS algorithm. This is inefficient especially in cryptography where large numbers are often tested for primality. Thus implementing AKS algorithm in MATLAB updated with MAPPLE is faster and efficient than using C++. 1. Introduction The study of prime numbers dates back to 300BC. An ancient Greek Mathematician called Euclid used the method of contradiction to prove that there were infinitely many prime numbers. He also proved the fundamental theorem of arithmetic. In 200BC, another Greek Mathematician called Eratosthenes introduced an algorithm for primality testing called the Sieve of Eratosthenes when given integer numbers from 2 to , mark all the even numbers except 2. Then start with the next unmarked number, which in our case is 3, mark all the numbers which are multiplies of three except 3. Continue in this way until there are no new unmarked numbers.
    [Show full text]
  • Improving the Speed and Accuracy of the Miller-Rabin Primality Test
    Improving the Speed and Accuracy of the Miller-Rabin Primality Test Shyam Narayanan Mentor: David Corwin MIT PRIMES-USA Abstract Currently, even the fastest deterministic primality tests run slowly, with the Agrawal- Kayal-Saxena (AKS) Primality Test runtime O~(log6(n)), and probabilistic primality tests such as the Fermat and Miller-Rabin Primality Tests are still prone to false re- sults. In this paper, we discuss the accuracy of the Miller-Rabin Primality Test and the number of nonwitnesses for a composite odd integer n. We also extend the Miller- '(n) Rabin Theorem by determining when the number of nonwitnesses N(n) equals 4 5 and by proving that for all n, if N(n) > · '(n) then n must be of one of these 3 32 forms: n = (2x + 1)(4x + 1), where x is an integer, n = (2x + 1)(6x + 1), where x is an integer, n is a Carmichael number of the form pqr, where p, q, r are distinct primes congruent to 3 (mod 4). We then find witnesses to certain forms of composite numbers with high rates of nonwitnesses and find that Jacobi nonresidues and 2 are both valuable bases for the Miller-Rabin test. Finally, we investigate the frequency of strong pseudoprimes further and analyze common patterns using MATLAB. This work is expected to result in a faster and better primality test for large integers. 1 1 Introduction Data is growing at an astoundingly rapid rate, and better information security is re- quired to protect increasing quantities of data. Improved data protection requires more sophisticated cryptographic methods.
    [Show full text]
  • Primality Testing : a Review
    Primality Testing : A Review Debdas Paul April 26, 2012 Abstract Primality testing was one of the greatest computational challenges for the the- oretical computer scientists and mathematicians until 2004 when Agrawal, Kayal and Saxena have proved that the problem belongs to complexity class P . This famous algorithm is named after the above three authors and now called The AKS Primality Testing having run time complexity O~(log10:5(n)). Further improvement has been done by Carl Pomerance and H. W. Lenstra Jr. by showing a variant of AKS primality test has running time O~(log7:5(n)). In this review, we discuss the gradual improvements in primality testing methods which lead to the breakthrough. We also discuss further improvements by Pomerance and Lenstra. 1 Introduction \The problem of distinguishing prime numbers from a composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such and extent that its would be superfluous to discuss the problem at length . Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated." - Johann Carl Friedrich Gauss, 1781 A number is a mathematical object which is used to count and measure any quantifiable object. The numbers which are developed to quantify natural objects are called Nat- ural Numbers. For example, 1; 2; 3; 4;::: . Further, we can partition the set of natural 1 numbers into three disjoint sets : f1g, set of fprime numbersg and set of fcomposite numbersg.
    [Show full text]
  • The Miller-Rabin Randomized Primality Test
    Introduction to Algorithms (CS 482) Cornell University Instructor: Bobby Kleinberg Lecture Notes, 5 May 2010 The Miller-Rabin Randomized Primality Test 1 Introduction Primality testing is an important algorithmic problem. In addition to being a fun- damental mathematical question, the problem of how to determine whether a given number is prime has tremendous practical importance. Every time someone uses the RSA public-key cryptosystem, they need to generate a private key consisting of two large prime numbers and a public key consisting of their product. To do this, one needs to be able to check rapidly whether a number is prime. The simplestp algorithm to test whether n is prime is trial division:p for k = 2; 3;:::; b nc test whether n ≡ 0 (mod k). This runs in time O( n log2(n)), but this running time is exponential in the input size since the input represents n as a binary number with dlog2(n)e digits. (A good public key these days relies on using prime numbers with at least 2250 binary digits; testing whether such a number is prime using trial division would require at least 2125 operations.) In 1980, Michael Rabin discovered a randomized polynomial-time algorithm to test whether a number is prime. It is called the Miller-Rabin primality test because it is closely related to a deterministic algorithm studied by Gary Miller in 1976. This is still the most practical known primality testing algorithm, and is widely used in software libraries that rely on RSA encryption, e.g. OpenSSL. 2 Randomized algorithms What does it mean to say that there is a randomized polynomial-time algorithm to solve a problem? Here are some definitions to make this notion precise.
    [Show full text]
  • Fibonacci's Answer to Primality Testing?
    FIBONACCI'S ANSWER TO PRIMALITY TESTING? JULIAN BEAUCHAMP Abstract. In this paper, we consider various approaches to primality testing and then ask whether an effective deterministic test for prime numbers can be found in the Fibonacci numbers. Introduction Prime numbers play an important role in encryption and cyber-security. They protect digitial data, including personal data and bank details. Testing whether a number is prime or not, is therefore becoming increasingly important for mathe- maticians and for those working in digitial security. The study of prime numbers and their properties go back to the ancient Greek mathematician Pythagoras (570-495 BC) who understood the idea of primality. But a primality test is a test to determine whether or not a number is prime. This is different from finding a number's constituent prime factors (also known as prime factorization). A number is said to be prime if it is divisible only by 1 and itself. Otherwise it is composite. When the numbers are small, it is relatively easy to determine whether a number is prime. But as they get exponentially larger they get harder to determine. So the pressing question is what makes an efficient algorithm? The following characteristics make an efficient algorithm - general, deterministic, unconditional, and polynomial:1 General. An algorithm that is general works for all numbers. Algorithms that are not general only work on certain numbers (e.g. the Lucas-Lehmer test for Mersenne numbers). Deterministic.A deterministic test (e.g. the Lucas-Lehmer Test) will tell us with absolute certainty whether a number is prime or not every time it is run.
    [Show full text]
  • Quantum Computation of Prime Number Functions
    Quantum Computation of Prime Number Functions Jos´eI. Latorre1 and Germ´anSierra2 1 Departament d'Estructura i Constituents de la Mat`eria,Universitat de Barcelona, Barcelona, Spain, Centre for Quantum Technologies, National University of Singapore, Singapore. 2 Instituto de F´ısicaTe´oricaUAM/CSIC, Universidad Aut´onomade Madrid, Cantoblanco, Madrid, Spain. We propose a quantum circuit that creates a pure state corresponding to the quantum super- position of all prime numbers less than 2n, where n is the number of qubits of the register. This Prime state can be built using Grover's algorithm, whose oracle is a quantum implementation of the classical Miller-Rabin primality test. The Prime state is highly entangled, and its entangle- ment measures encode number theoretical functions such as the distribution of twin primes or the Chebyshev bias. This algorithm can be further combined with the quantum Fourier transform to yield an estimate of the prime counting function, more efficiently than any classical algorithm and with an error below the bound that allows for the verification of the Riemann hypothesis. We also propose a Twin Prime state to measure the number of twin primes and another state to test the Goldbach conjecture. Arithmetic properties of prime numbers are then, in principle, amenable to experimental verifications on quantum systems. I. INTRODUCTION Prime numbers are central objects in Mathematics and Computer Science. They appeared dramatically in Quantum Computation through the Shor's algorithm, which converts the hard problem of factorization into a polynomial one using quantum interference1,2. In Number Theory, prime numbers are fully characterized by the prime counting function π(x), which is the number of primes less or equal to x.
    [Show full text]
  • Primality Testing, Randomness & Derandomization
    Primality Testing, Randomness & Derandomization Nitin Saxena Centrum voor Wiskunde en Informatica Amsterdam Bonn, May 2007 Nitin Saxena (CWI, Amsterdam) Derandomization Bonn, May 2007 1 / 41 Outline 1 Prologue 2 Randomness 3 Derandomizing Primality Testing 4 Derandomizing Identity Testing Depth 3 Circuits: Algorithm I Depth 3 Circuits: Algorithm II 5 Epilogue Nitin Saxena (CWI, Amsterdam) Derandomization Bonn, May 2007 2 / 41 Prologue Outline 1 Prologue 2 Randomness 3 Derandomizing Primality Testing 4 Derandomizing Identity Testing Depth 3 Circuits: Algorithm I Depth 3 Circuits: Algorithm II 5 Epilogue Nitin Saxena (CWI, Amsterdam) Derandomization Bonn, May 2007 3 / 41 Prologue Primality: The Problem Given an integer n, test whether it is prime. Easy Solution: Divide n by all numbers between 2 and (n − 1). But we would like to do this in time polynomial in log n. First asked by Kurt G¨odel in a letter to John von Neumann in 1956. Nitin Saxena (CWI, Amsterdam) Derandomization Bonn, May 2007 4 / 41 Prologue Primality: The Problem Given an integer n, test whether it is prime. Easy Solution: Divide n by all numbers between 2 and (n − 1). But we would like to do this in time polynomial in log n. First asked by Kurt G¨odel in a letter to John von Neumann in 1956. Nitin Saxena (CWI, Amsterdam) Derandomization Bonn, May 2007 4 / 41 Prologue Primality: The Problem Given an integer n, test whether it is prime. Easy Solution: Divide n by all numbers between 2 and (n − 1). But we would like to do this in time polynomial in log n.
    [Show full text]
  • An Algorithm That Decides PRIMES in Polynomial Time
    An Algorithm that Decides PRIMES in Polynomial Time Kevin Clark June 3, 2011 Contents 1 Introduction 2 2 Definitions 2 3 Description of the Algorithm 3 4 Correctness of the Algorithm 7 5 Asymptotic Analysis of the Algorithm 7 1 1 Introduction Prime numbers have been studied by mathematicians since the time of Pythagoras. They play an important role in many areas of mathematics, especially in number theory. One important question arising from the study of prime number is if the primality of a number can be determined efficiently. This has practical as well as theoretical value because modern cryptography relies on generating large prime numbers. Since the advent of complexity theory, which allowed the idea of ”efficiency” to be for- mulated precisely, this problem has been studied intensively. Primality was known to be in NP (solvable in nondeterministic polynomial time) since 1974 but it remained an open question until 2003 whether it was in P (solvable in polynomial time). A simple algorithm for deciding if n is prime called the Sievep of Erasthones tests if n can be divided by any natural number less than or equal to n. This algorithm runs in exponential time on its input, giving rise to the question of whether a faster algorithm for deciding primes exists. A series of breakthroughs brought algorithms closer to deterministically deciding primality in polynomial time. A probabilistic algorithm by Miller and Rabin [7] relying on Fermat's Little Theorem can find prime numbers in polynomial time given a probability that the algorithm produces incorrect output. In 1983, Adleman, Pomerance, and Rumely [1] gave a deterministic algorithm that runs in Olog nlog log log n time, which is better than exponential time (although still not polynomial).
    [Show full text]
  • Primality Tests
    Primality Tests Andreas Klappenecker Texas A&M University © 2018 by Andreas Klappenecker. All rights reserved. 1 / 30 Question Suppose that Bob chooses the large number such as n “ 456 989 977 669 How can Bob check whether n is prime? Actually, this is a \small number". We are usually interested in testing primality of numbers with hundreds of digits, but those do not look too nice on a slide. 2 / 30 AKS Primality Test Use the Agrawal-Kayal-Saxena primality test. It is a deterministic Oplogpnq12q time algorithm. Space requirements make the test impractical for large n. r Unlike integer factorization into primes, we know a poly-time algorithm for primality testing, but it is not too useful in practice. 3 / 30 Goal Goal We will now develop some randomized algorithms for primality testing. 4 / 30 Fermat 5 / 30 Fermat's Little Theorem We need the following simple result from number theory. Fermat's Little Theorem Let p be a prime. Then ap ” a pmod pq for all integers a. 6 / 30 Proof of Fermat's Little Theorem (1/2) Case 1. Suppose that p divides a. Then a ” 0 ” ap pmod pq. Case 2. Suppose that p does not divide a. Then ap ” a pmod pq is equivalent to ap´1 ” 1 pmod pq. Consider the p ´ 1 numbers a; 2a; 3a;:::; pp ´ 1qa: We claim that they are all different mod p. Indeed, if we would have ja ” ka pmod pq, then pj ´ kqa ” 0 pmod pq. Since a ı 0 pmod pq, we must have pj ´ kq ” 0 pmod pq.
    [Show full text]