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Problems

``I think it's very important that people are encouraged to work on very hard problems. The tendency today is to work on short and immediate problems." This is attributed to Andrew Wiles.

The Pomerance 10

On Saturday, June 10th, 2000, Carl Pomerance gave a plenary lecture at the Canadian Mathematical Society summer meeting at McMaster University, Hamilton, Ontario, Canada. He listed his top 10 unsolved problems in number theory.⁰¹⁵ I formulated the problems and so I am accountable for any mistakes. The numbering of my list does not necessarily agree with the numbering of Carl's list. -Ed.

Goldbach Conjecture. In a letter to Euler in 1742, Goldbach formulated the following conjecture. Conjecture (Goldbach) Every even integer A $\geqslant$ 4 is the sum of two primes.
Riemann Conjecture. For a complex number s = $\sigma$ + it, and $\chi$ (n) a non-trivial Dirichlet character, let

$\displaystyle \zeta$ (s) = $\displaystyle \sum_{n\geqslant 1}^{}$ $\displaystyle {\frac{1}{n^s}}$ ,
and

L(s, $\displaystyle \chi$ ) = $\displaystyle \sum_{n\geqslant 1}^{}$ $\displaystyle {\frac{\chi(n)}{n^s}}$ .
Conjecture (Riemann Hypothesis-RH) All non-trivial zeros of $\zeta$ (s) lie on the line $\sigma$ = ${\frac{1}{2}}$ . Conjecture (Extended Riemann Hypothesis-ERH) All non-trivial zeros of L(s, $\chi$ ) lie on the line $\sigma$ = ${\frac{1}{2}}$ . Let K be an algebraic number field, $\mathcal {O}$ _K its ring of integers, $\bf\it a$ an ideal of $\mathcal {O}$ _K, and

$\displaystyle \zeta_{K}^{}$ (s) = $\displaystyle \sum_{{\bf\it a}}^{}$ $\displaystyle {\frac{1}{(\mathrm{N}{\bf\it a})^s}}$ ,
where the sum is over all ideals $\bf\it a$ of $\mathcal {O}$ _K, and N $\bf\it a$ is the norm of $\bf\it a$ , defined to be the index of $\bf\it a$ in $\mathcal {O}$ _K. Conjecture (Generalised Riemann Hypothesis-GRH) All non-trivial zeros of $\zeta_{K}^{}$ (s) lie on the line $\sigma$ = ${\frac{1}{2}}$ .
Hardy-Littlewood Conjecture. For integers A $\neq$ 0, B, and C, let

q(n) = An² + Bn + C, for some n,
and

Q(x) = #{ p $\displaystyle \leqslant$ x : p = q(n) for some n }.
Conjecture (Hardy-Littlewood, 1923). There exists a constant K > 0 such that

Q(x) $\displaystyle \thicksim$ K $\displaystyle {\frac{\sqrt x}{\log x}}$ .
Artin's Conjecture. Let A be a non-zero integer other than 1, - 1, or a perfect square, and let N_A(x) denote the number of primes p $\leqslant$ x for which A generates the cyclic group $\mathbb {F}$ _p^* = { 1, 2, 3,..., p - 1 }. Conjecture (Artin, 1927) There exists a constant K(A) > 0 such that,

N_A(x) $\displaystyle \thicksim$ K(A) $\displaystyle {\frac{\sqrt x}{\log x}}$ .
ABC Conjecture. In 1980, Masser and Oesterlé formulated the following conjecture. Conjecture (Masser-Oesterlé, 1980) Let n = p₁p₂^...p_k, and rad (n) $\;\stackrel{\mathrm{def}}{=}\;$ p₁p₂^...p_k. Suppose we have 3 integers A, B, C satisfying (A, B) = (A, C) = (B, C) = 1, and A + B = C. Given $\epsilon$ > 0, it is conjectured that there is a constant K( $\epsilon$ ) such that

max(| A|,| B|,| C|) $\displaystyle \leqslant$ K( $\displaystyle \epsilon$ ){rad (ABC)}^{1 +}.
Discrete Log Problem. Let G be a finite group with group operation o, $\alpha$ $\in$ G, H = { $\alpha^{i}_{}$ : i $\geqslant$ 0 } the subgroup generated by $\alpha$ , and $\beta$ $\in$ H. Discrete Log ProblemFind the unique integer A such that 0 $\leqslant$ A $\leqslant$ #H - 1 and $\alpha^{A}_{}$ = $\beta$ , where $\alpha^{A}_{}$ means $\alpha$ o $\alpha$ o^...o $\alpha$ o $\alpha$ , A times.
Wieferich Primes. An odd prime p is called a Wieferich prime if p² divides 2^{p - 1} - 1. The only known Wieferich primes are 1093 and 3511. Wieferich Prime Problem Do there exist infinitely many primes p such that p² divides 2^{p - 1} - 1?
Mersenne Primes. A Mersenne prime number is a prime of the form 2^p - 1. Notice every even perfect number; equal to the sum of its proper divisors, is of the form 2^{p - 1}(2^p - 1); 6 = 2(2² - 1), 28 = 2²(2³ - 1), 496 = 2⁴(2⁵ - 1),... Mersenne Prime Problem Do there exist infinitely many primes p such that 2^p - 1 is a prime? That is, do there exist infinitely many even perfect numbers? Conjecture (Wagstaff, 1983) There exists a constant K > 0 such that

#{ p $\displaystyle \leqslant$ x : 2^p - 1 is prime } $\displaystyle \thicksim$ K loglog x.
Prime Gaps. Let p_n denote the nth prime, and let d_n = p_{n + 1} - p_n. Prime Gap Problem For p_{n + 1} $\leqslant$ x, how big can d_n be?
Multiplication vs. Factoring. Modern cryptography is based on our inability to factor large numbers fast. Arithemetic Problem Prove or disprove the following assertion which is at the base of the RSA cryposystem: It is harder to factor a given number than to multiply given numbers together.

Remarks

On the Riemann Hypothesis. Show that

$\displaystyle \sum_{d\vert n}^{}$ $\displaystyle {\frac{1}{d}}$ < eloglog n,
for all n sufficiently large ( n $\geqslant$ 5041), where $\gamma$ is Euler's constant. Theorem (Robin, G, 1984): This is equivalent to RH. A. Ivic⁰¹⁶ has shown for n $\geqslant$ 7, $\sum_{d\vert n}^{}$ ${\frac{1}{d}}$ < 2.59 loglog n. Let

$\displaystyle \theta_{n,m}^{}$ = $\displaystyle \sum_{k_1<\cdots<k_n}^{}$ $\displaystyle \prod_{\ell=1}^{n}$ [ $\displaystyle \rho_{k_\ell}^{}$ (1 - $\displaystyle \rho_{k_\ell}^{}$ )]^-m,
where $\rho_{1}^{}$ , $\rho_{2}^{}$ ,^... is the list of all nontrivial zeros of $\zeta$ (s). Theorem (Matijaševic, 1988): For all n, m, $\theta_{n,m}^{}$ > 0 is equivalent to RH.
On the Extended Riemann Hypothesis. Theorem (Miller, G, 1975): ERH implies that primality testing can be done in polynomial time P. Is primality testing NP complete? If so, ERH implies P=NP.
On the Generalized Riemann Hypothesis. Theorem (Hooley, C, 1967): GRH implies Artin's conjecture.
Hilbert's problem 8 (in 3 parts and concerned with RH, Goldbach's conjecture and twin primes) is still unresolved since its announcement as part of Hilbert's famous list of 23 problems. Notice that Hilbert's list of 23 problems appears in the Bulletin (New Series) of the American Mathematical Society (AMS) volume 37, #4, October 2000, pg. 407.

The Riemann Hypothesis has been supported by extensive numerical calculations (the first 1.5 billion zeros satisfy RH) and various heuristics, but such partial evidence is not convincing enough for all mathematicians, including A. Ivic. He points out⁰¹⁷ a heuristic connected with moments of the zeta-functions: this would imply the falsity of the Lindelöf hypothesis, and a fortiori that of the Riemann hypothesis! In the same paper he discusses other ``conditional disproofs" of RH. Furthermore, in personal correspondence, he writes ``I don't think there is much hope in working through Robin's equivalent."

CMI Millennium Prize Problems

The Clay Mathematics Institute (CMI) of Cambridge, Massachusetts has named seven Millennium Prize Problems. The Scientific Advisory Board of CMI selected these problems, focusing on important classic questions that have resisted solution over the years. The Board of Directors of CMI designated a 7 million prize fund for the solution to these problems, with 1 million allocated to each. The problems were announced on Wednesday, May 24, 2000, at the Collège de France. One hundred years earlier, on August 8, 1900, David Hilbert delivered his famous 23 open problems. A mathematical description (a leading specialist in the domain in question has formulated each problem) as well as the rules for the CMI millennium prize problems can be found via our website (http://www.cms.math.ca/Students) by following the links In your area, mathematics at UNBC, news, Clay Mathematics. The 7 problems are:

P versus NP

The Hodge Conjecture

The Poincaré Conjecture

The Riemann Hypothesis

Yang-Mills Existence and Mass Gap

Navier-Stokes Existence and Smoothness

The Birch and Swinnerton-Dyer Conjecture

Andrew J. Irwin 2001-03-19

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