## Visions of Infinity: The Great Mathematical ProblemsIt is one of the wonders of mathematics that, for every problem mathematicians solve, another awaits to perplex and galvanize them. Some of these problems are new, while others have puzzled and bewitched thinkers across the ages. Such challenges offer a tantalizing glimpse of the field's unlimited potential, and keep mathematicians looking toward the horizons of intellectual possibility. In Visions of Infinity, celebrated mathematician Ian Stewart provides a fascinating overview of the most formidable problems mathematicians have vanquished, and those that vex them still. He explains why these problems exist, what drives mathematicians to solve them, and why their efforts matter in the context of science as a whole. The three-century effort to prove Fermat's last theorem—first posited in 1630, and finally solved by Andrew Wiles in 1995—led to the creation of algebraic number theory and complex analysis. The Poincaré conjecture, which was cracked in 2002 by the eccentric genius Grigori Perelman, has become fundamental to mathematicians' understanding of three-dimensional shapes. But while mathematicians have made enormous advances in recent years, some problems continue to baffle us. Indeed, the Riemann hypothesis, which Stewart refers to as the “Holy Grail of pure mathematics,” and the P/NP problem, which straddles mathematics and computer science, could easily remain unproved for another hundred years.An approachable and illuminating history of mathematics as told through fourteen of its greatest problems, Visions of Infinity reveals how mathematicians the world over are rising to the challenges set by their predecessors—and how the enigmas of the past inevitably surrender to the powerful techniques of the present. |

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#### LibraryThing Review

User Review - amarcobio - LibraryThingI really feel that my opinion of Ian Stewart has evolved. I found the first books I read not particularly good, while the recent one more interesting. In any case I frequently browse my local book ... Read full review

#### LibraryThing Review

User Review - fpagan - LibraryThingAccomplished math popularizer Stewart here discusses 14 longstanding problems that have either been solved relatively recently (e.g. 4-color, Fermat's Last Theorem) or are still unsolved (e.g. P=NP ... Read full review

### Contents

Prime territory I Goldbach Conjecture | 16 |

The puzzle of pi I Squaring the Circle | 40 |

Mapmaking mysteries I Four Colour Theorem | 58 |

Sphereful symmetry I Kepler Conjecture | 80 |

New solutions for old I Mordell Conjecture | 102 |

Inadequate margins I Fermats Last Theorem | 115 |

Orbital chaos I ThreeBody Problem | 136 |

Patterns in primes I Riemann Hypothesis | 153 |

They cant all be easy I PNP Problem | 203 |

Quantum conundrum I Mass Gap Hypothesis | 228 |

Diophantine dreams I BirchSWinnertonDyer Conjecture | 245 |

Complex cycles I Hodge Conjecture | 258 |

Where next? | 277 |

Glossary | 294 |

Notes | 307 |

325 | |

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### Common terms and phrases

3-sphere ABC conjecture algebraic number algorithm angle answer area of mathematics arithmetic atoms calculation chapter circle coefﬁcients complex numbers conﬁgurations congruent construction curvature cycle deﬁned deﬁnition difﬁcult dimensions Diophantine equation edges efﬁcient elliptic curve Euclid’s Euler example exist factorisation Fermat’s last theorem ﬁgure ﬁnd ﬁnding ﬁnite ﬁrst ﬁt ﬁve ﬁxed ﬂat ﬂuid formula four colour four colour problem four colour theorem Gauss generalised geometry Hodge conjecture homology idea inﬁnitely integer Kepler conjecture known lattice layer loops mathematicians mathematics methods minimal criminal modular multiple Navier-Stokes equation neighbours number theory orbit packing particles plane Poincare Poincare conjecture polygon polynomial prime factors prime number prime number theorem problem proof proved Pythagorean triples question real numbers region result Ricci ﬂow Riemann hypothesis signiﬁcant solve space speciﬁc sphere square sufﬁciently surface symmetries there’s topological torus triangle trip variables whole numbers zero zeta function