XM: Introduction. Søren Eilers Institut for Matematiske Fag KØBENHAVNS UNIVERSITET I N S T I T U T F O R M A T E M A T I S K E F A G
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1 I N S T I T U T F O R M A T E M A T I S K E F A G KØBENHAVNS UNIVERSITET XM: Introduction Søren Eilers Institut for Matematiske Fag November 17, 2016 Dias 1/19
2 Outline 1 What is XM? 2 Historical experiments 3 The XM toolbox Dias 2/19 XM: Introduction November 17, 2016
3 1.2 Basic methodology 7 Initial investigation Common features in examples Initial hypothesis Experiment, search for counterexamples Counterexamples Common features in counterexamples Revised hypothesis Experiment, search for counterexamples Proof Knowledge gained from the experiment Final hypothesis Common features in counterexamples Figure 1.3 Stepwise refinement of hypotheses through a systematic search for counterexamples. Dias 3/19 XM: Introduction November 17, 2016 checked). Nevertheless, it is very useful for the experimenting mathematician to employ a similar strategy in the investigation of mathematical objects. An
4 Experimental mathematics The systematic investigation of concrete examples of a mathematical structure in the search for conjectures about its properties (using computers). Dias 4/19 XM: Introduction November 17, 2016
5 Experimental mathematics The systematic investigation of concrete examples of a mathematical structure in the search for conjectures about its properties (using computers). Typical output from an experimental investigation: A counterexample Dias 4/19 XM: Introduction November 17, 2016
6 Experimental mathematics The systematic investigation of concrete examples of a mathematical structure in the search for conjectures about its properties (using computers). Typical output from an experimental investigation: A counterexample A reference to the mathematical literature Dias 4/19 XM: Introduction November 17, 2016
7 Experimental mathematics The systematic investigation of concrete examples of a mathematical structure in the search for conjectures about its properties (using computers). Typical output from an experimental investigation: A counterexample A reference to the mathematical literature A hint for a proof Dias 4/19 XM: Introduction November 17, 2016
8 Experimental mathematics The systematic investigation of concrete examples of a mathematical structure in the search for conjectures about its properties (using computers). Typical output from an experimental investigation: A counterexample A reference to the mathematical literature A hint for a proof A conjecture Dias 4/19 XM: Introduction November 17, 2016
9 Conjecture [L. Euler 1769] x n 1 ` x n 2 ` ` x n m y n can only have solutions x 1,x 2,...,x m,y P N when n ě m for m ě 2; i.e., for a sum of nth powers to itself be an nth power it must have either only one or at least n summands. Dias 5/19 XM: Introduction November 17, 2016
10 Conjecture [L. Euler 1769] x n 1 ` x n 2 ` ` x n m y n can only have solutions x 1,x 2,...,x m,y P N when n ě m for m ě 2; i.e., for a sum of nth powers to itself be an nth power it must have either only one or at least n summands. Euler had essentially proved that x 3 1 ` x 3 2 y 3 did not have any such solutions, cf. Fermat s last theorem. Dias 5/19 XM: Introduction November 17, 2016
11 30 (1930), K Ø B E N H A V N S U N I V E R S I T E T I N S T I T U T F O R M A T E M A T I S K E F A G DARTMOUTH COLLEGE COUNTEREXAMPLE TO EULER'S CONJECTURE ON SUMS OF LIKE POWERS BY L. J. LANDER AND T. R. PARKIN Communicated by J. D. Swift, June 27, 1966 A direct search on the CDC 6600 yielded HO as the smallest instance in which four fifth powers sum to a fifth power. This is a counterexample to a conjecture by Euler [l] that at least n nth powers are required to sum to an nth power, n>2. REFERENCE 1. L. E. Dickson, History of the theory of numbers, Vol. 2, Chelsea, New York, 1952, p Dias 6/19 XM: Introduction November 17, 2016
12 Birch & Swinnerton-Dyer: Data for the elliptic curves y 2 x 3 d 2 x at d 1, 5, 34,1254, Dias 7/19 XM: Introduction November 17, 2016
13 Consider an elliptic curve E : y 2 x 3 ` ax ` b, a,b P Z with discriminant 16p4a 3 ` 27b 2 q 0. For every prime p not dividing, consider N p 1 ` ˇˇt0 ď x,y ď p 1 y 2 x 3 ` ax ` b mod puˇˇ. Birch and Swinnerton-Dyer got the idea that N p should be related to the rank of E, and over a period of five years in the late 1950 s and early 1960 s they computed π E pxq ź pďx,p for many different values of X for a number of elliptic curves with known ranks, leading to the conjecture that N p p π E pxq Ñ C lnpxq rankpeq for X Ñ 8 for some C that depends only on E. Dias 8/19 XM: Introduction November 17, 2016
14 1.2 Basic methodology 7 Initial investigation Common features in examples Initial hypothesis Experiment, search for counterexamples Counterexamples Common features in counterexamples Revised hypothesis Experiment, search for counterexamples Proof Knowledge gained from the experiment Final hypothesis Common features in counterexamples Figure 1.3 Stepwise refinement of hypotheses through a systematic search for counterexamples. Dias 9/19 XM: Introduction November 17, 2016 checked). Nevertheless, it is very useful for the experimenting mathematician to employ a similar strategy in the investigation of mathematical objects. An
15 Outline 1 What is XM? 2 Historical experiments 3 The XM toolbox Dias 10/19 XM: Introduction November 17, 2016
16 2 1 ` 1 3 is a prime 2 2 ` 1 5 is a prime 2 4 ` 1 17 is a prime 2 8 ` is a prime 2 16 ` is a prime Dias 11/19 XM: Introduction November 17, 2016
17 2 1 ` 1 3 is a prime 2 2 ` 1 5 is a prime 2 4 ` 1 17 is a prime 2 8 ` is a prime 2 16 ` is a prime Lemma If 2 m ` 1 is a prime, then m 2 k Dias 11/19 XM: Introduction November 17, 2016
18 2 1 ` 1 3 is a prime 2 2 ` 1 5 is a prime 2 4 ` 1 17 is a prime 2 8 ` is a prime 2 16 ` is a prime Lemma If 2 m ` 1 is a prime, then m 2 k Conjecture [P. de Fermat] Every number F m 2 2m ` 1 is prime Dias 11/19 XM: Introduction November 17, 2016
19 2 1 ` 1 3 is a prime 2 2 ` 1 5 is a prime 2 4 ` 1 17 is a prime 2 8 ` is a prime 2 16 ` is a prime Lemma If 2 m ` 1 is a prime, then m 2 k Conjecture [P. de Fermat] Every number F m 2 2m ` 1 is prime Example [L. Euler 1732] F Dias 11/19 XM: Introduction November 17, 2016
20 Dias 12/19 XM: Introduction November 17, 2016
21 Conjecture [C.F. Gauss, ] With πpxq the number of primes less than x, πpxq «ż x 2 du lnu : Lipxq Dias 13/19 XM: Introduction November 17, 2016
22 Conjecture [C.F. Gauss, ] With πpxq the number of primes less than x, πpxq «ż x 2 du lnu : Lipxq The prime number theorem [J. Hadamard, P. de la Vallée-Poussin 1896] πpxq lim xñ8 Lipxq 1 Dias 13/19 XM: Introduction November 17, 2016
23 Outline 1 What is XM? 2 Historical experiments 3 The XM toolbox Dias 14/19 XM: Introduction November 17, 2016
24 1 Introduction 2 Basic programming in Maple 3 Iteration and recursion 4 Visualization 5 Symbolic inversion 6 Pseudorandomness 7 Time, memory and precision 8 Linear algebra and graph theory Dias 15/19 XM: Introduction November 17, 2016
25 1 Introduction and basic programming in Maple 2 Iteration and recursion 3 Visualization 4 Symbolic inversion 5 Pseudorandomness 6 Time, memory and precision 7 Linear algebra and graph theory Dias 16/19 XM: Introduction November 17, 2016
26 1 Introduction and basic programming in Maple [ALG] 2 Iteration and recursion [ALG] 3 Visualization 4 Symbolic inversion 5 Pseudorandomness 6 Time, memory and precision [ALG] 7 Linear algebra and graph theory Dias 16/19 XM: Introduction November 17, 2016
27 2.6 Pseudocode and stepwise refinement 41 Example : Sieve of Eratosthenes. The Sieve of Eratosthenes is a simple algorithm for exhaustively finding the primes less than or equal to a given upper bound N. Pseudocode for the sieve is given in Algorithm 2.1. Algorithm 2.1 Sieve of Eratosthenes 1: procedure Sieve(N) 2: Construct a list L = [2,..., N] 3: Let p = 2 4: while p < N do 5: Remove ip from L for each integer i 2 6: Let p be the smallest element in L greater than p 7: end while 8: return L L contains all primes smaller than N 9: end procedure Dias 17/19 XM: Introduction November 17, 2016 Algorithm 2.2 Stepwise refinement 1: identify the problem that you want to solve 2: write initial pseudocode for the program 3: while the pseudocode is not detailed enough to write the program do 4: modify and refine the pseudocode 5: end while
28 1 Introduction and basic programming in Maple [ALG] 2 Iteration and recursion [ALG] 3 Visualization 4 Symbolic inversion 5 Pseudorandomness 6 Time, memory and precision [ALG] 7 Linear algebra and graph theory Dias 18/19 XM: Introduction November 17, 2016
29 1 Introduction and basic programming in Maple [ALG] 2 Iteration and recursion [ALG] 3 Visualization [VIS] 4 Symbolic inversion 5 Pseudorandomness 6 Time, memory and precision [ALG] 7 Linear algebra and graph theory Dias 18/19 XM: Introduction November 17, 2016
30 1 Introduction and basic programming in Maple [ALG] 2 Iteration and recursion [ALG] 3 Visualization [VIS] 4 Symbolic inversion 5 Pseudorandomness [RAN] 6 Time, memory and precision [ALG] 7 Linear algebra and graph theory Dias 18/19 XM: Introduction November 17, 2016
31 1 Introduction and basic programming in Maple [ALG] 2 Iteration and recursion [ALG] 3 Visualization [VIS] 4 Symbolic inversion [INV] 5 Pseudorandomness [RAN] 6 Time, memory and precision [ALG] 7 Linear algebra and graph theory Dias 18/19 XM: Introduction November 17, 2016
32 Question What is ? Question What is 1,2,5,12,32,94,289,910? Dias 19/19 XM: Introduction November 17, 2016
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