Appendix A Famous Conjectures in Mathematics There are many perplexing questions in mathematics that have not been resolved. For most of these open questions—and particularly since computing power has increased—researchers have been able to guess the answer; the question is not con- sidered settled, however, until a rigorous proof has been established. A conjecture is a statement for which there is no proof yet known. While conjectures turn into theorems on a daily basis, one suspects that, as mathematicians are introducing and investigating new concepts in an ever-growing number of fields, the number of unsolved questions in mathematics is actually increasing. Even about such familiar concepts as the positive integers, there is a lot we do not know. Here is our selection for the top ten most famous conjectures about the integers, together with some information on how much we currently know about them. Odd Perfect Number Conjecture. There are no odd perfect numbers. (It has been verified that there are no odd perfect numbers under 10300; see also Chap. 3.) Conjecture on Prime Values of Polynomials. There are polynomials of one variable and of degree two or more that assume infinitely many prime values. In particular, there are infinitely many values of n for which n2 C 1 and n2 n C 41 are prime numbers. (It has been known that there are infinitely many values of n for which n2 C 1 is either prime or the product of two primes; the same holds for n2 n C 41. The polynomial n2 n C 41 generates primes for all 39 Ä n Ä 40; see Chap. 3.) Waring’s Problem Conjecture. The smallest integer m for which every positive integer n can be expressed as the sum of m terms, each of which is a perfect k-th power, is $Â Ã % 3 k g.k/ D 2k C 2: 2 (At the present time we know that this holds for every k Ä 471; 600; 000 and every sufficiently large k, thus leaving only finitely many cases open; see Chap. 3). B. Bajnok, An Invitation to Abstract Mathematics, Undergraduate Texts in Mathematics, 381 DOI 10.1007/978-1-4614-6636-9, © Bela´ Bajnok 2013 382 A Famous Conjectures in Mathematics The Mersenne Prime Conjecture. There are infinitely positive integers n for which 2n 1 is a prime number. (There are currently 47 Mersenne primes known; see Chap. 2.) The Fermat Prime Conjecture. There are only five nonnegative integers n for which 2n C 1 is a prime number. (It has been shown that no other n yields prime up to n D 32; see Chap. 2.) The Euclid Number Conjecture. There are infinitely many positive integers n for which Euclid’s number Yn Kn D 1 C pi iD1 is a prime number. (It has been shown that the only values of n under 200 for which Kn is prime are n D 1; 2; 3; 4; 5; 11; 75; 171; and 172; see also Chaps. 3 and 5.) The Twin Prime Conjecture. There are infinitely many pairs of prime numbers that differ by 2. (The largest twin primes currently known have more than two hundred thousand digits; see Chap. 5.) The Goldbach Conjecture. Every even integer that is greater than 2 can be expressed as the sum of two positive prime numbers. (As of today, the Goldbach Conjecture has been shown to be true for all even integers up to about 1018.Note that, as a consequence, every odd integer that is greater than 5 can be expressed as the sum of three positive prime numbers; this latter claim is also unknown and is usually referred to as the Weak Goldbach Conjecture. The Weak Goldbach Conjecture has been proved for all odd integers greater than about 2 101346, leaving only finitely many cases open that, in theory, can be completed with the help of improved computer technology.) Beal’s Conjecture. If a; b,andc are relatively prime positive integers, then the equation ax C by D cz has no positive integer solutions with x, y,andz all greater than 2. (The banker and amateur mathematician Andrew Beal offered $100,000 for a solution of this conjecture; the funds are held in trust by the American Mathematical Society.) The Collatz 3x C 1 Conjecture. On the positive integers, define the function x f.x/ D 3x C 1 if x is odd and f.x/ D 2 if x is even. Then the iteration of f starting with any initial value a (i.e., the sequence a; f .a/; f .f .a//; : : : ) eventually leads to 1. (This conjecture is currently verified for all initial values up to 5:6 1013.) We have to admit that, in our selection of these conjectures, we had a preference for those questions that could be stated without complicated technicalities. The questions considered most important, however, often require background that would take more time to explain. In the year 1900, the German mathematician David Hilbert (1862–1943), at the second International Congress of Mathematicians held in Paris, stated 23 open questions ranging over most branches of mathematics. (Many of Hilbert’s Problems, as they became known, are still open. As it turns out, Hilbert was mistaken in thinking that every well-phrased problem has a solution: A Famous Conjectures in Mathematics 383 the first problem on his list, the so-called Continuum Hypothesis, is now known to be independent of the usual axioms of mathematics; see Chap. 22.) One hundred years later, at the turn of the twenty-first century, several mathemati- cians collected what they believed were the most important open questions. Seven of these questions are known as the Millennium Problems; the Clay Mathematics Institute offers one-million-dollar prizes for the solution of each of seven important conjectures (e.g., the Riemann Hypothesis, the Poincare´ Conjecture, and “P D NP”)—see www.claymath.org for more information. As of now, six of the Millennium Problems remain open; the Poincare´ Conjecture, however, was solved in 2003 in a fantastic achievement by Grigori Perelman of the Steklov Institute of Mathematics in St. Petersburg. (Perelman chose not to accept the Millennium Prize, as he also declined the most prestigious award in mathematics, the Fields Medal. As he told the President of the International Mathematical Union, the prize “was completely irrelevant for me. Everybody understood that if the proof is correct then no other recognition is needed.”) The Poincare´ Conjecture deals with the characterization of n-dimensional spheres, a fundamental topic in topology. We attempt to explain the Poincare´ Conjecture as follows. In topology, one considers two shapes equivalent if each one can be continuously transformed into the other—breaking and punching holes are not allowed. For example, a topologist would say that a doughnut and a coffee cup are equivalent, but an apple is different. One way to see the difference is by considering various loops on these surfaces. Every loop on an apple can be continuously moved to any other loop—this, however, cannot be said about the coffee cup where, for example, a loop around the ear of the cup cannot be moved to a loop on the side (without breaking up the loop). In 1904, the French mathematician Henri Poincare´ asked whether this simple loop test is enough to identify S n,the n-dimensional sphere. Using more precise (yet here undefined) terminology, the Poincare´ Conjecture (now theorem) can be stated as follows: Theorem A.1 (The PoincareTheorem).´ Every compact, simply connected, smooth n-dimensional manifold is equivalent to the sphere S n. The two-dimensional case was quickly solved by Poincare´ himself, but several decades passed before the next breakthrough occurred. In 1961, Stephen Smale proved the conjecture for every n 5; then, in 1982, Michael Freedman resolved the case n D 4. This left only the three-dimensional case open, which became the non plus ultra question in topology. Perelman’s breakthrough came after deep and extensive work done by many mathematicians. The assignments below offer you a wide variety of project possibilities for further investigation of these conjectures and results as well as additional questions that await mathematicians of the future. 384 A Famous Conjectures in Mathematics Assignments 1. Recall that we call a positive integer n perfect if its positive divisors other than itself add up to n. Related to perfect numbers, we define a pair of positive integers m and n amicable if the positive divisors of m other than m add up to n and the positive divisors of n other than n adduptom. As of today, over 10 million pairs of amicable numbers are known, and it is a famous old conjecture that there are infinitely many. (a) Verify that 220 and 284 form an amicable pair. (b) There have been numerous rules discovered that, under certain conditions, yield amicable numbers. One such rule, known since the tenth century, k kC1 2kC1 states that if p1 D 3 2 1, p2 D 3 2 1,andp3 D 9 2 1 are kC1 kC1 primes for some positive integer k,thenm D 2 p1 p2 and n D 2 p3 are amicable. (For k D 1 we get m D 220 and n D 284.) Verify this result. 2. Here we investigate prime numbers in arithmetic progressions, that is, a sequence of prime numbers of the form a; a C d;a C 2d;:::;aC kd for some positive integers a, d,andk. (a) Find three primes in arithmetic progression. (b) Find four primes in arithmetic progression. (c) Find five primes in arithmetic progression.