08-17-2026, 05:33 PM
Unsolved Problems in Number Theory
Author: Richard K. Guy
Publisher: Springer
Publication: 3rd edition, 2004
Series:Problem Books in Mathematics
Richard K. Guy’s Unsolved Problems in Number Theory is a classic reference devoted not to teaching number theory systematically, but to showing where our knowledge of integers reaches its limits. Guy assembles a remarkably large collection of open questions, conjectures, partial results, computational observations, and references. Many problems are extremely easy to state yet extraordinarily difficult to solve—one of the characteristic attractions of number theory. The book is organized around six broad areas: prime numbers, divisibility, additive number theory, Diophantine equations, integer sequences, and miscellaneous problems.
The prime-number sections include questions concerning prime-producing formulas, Mersenne and Fermat numbers, prime gaps, arithmetic progressions of primes, pseudoprimes, and patterns among consecutive primes. Other chapters explore divisibility properties, representations of integers as sums, Diophantine equations, and unusual integer sequences. Rather than simply presenting a catalogue of puzzles, Guy explains what was known about each question, records related results and conjectures, and points readers toward the mathematical literature. Consequently, the book can function as a map of research problems: a reader can start with an elementary-looking question and quickly discover the deeper mathematics surrounding it.
The third edition substantially expanded the earlier versions, adding problems involving symmetric and asymmetric primes, sums of higher powers, Diophantine $m$-tuples, Conway's RATS and palindromes, together with references to the OEIS (Online Encyclopedia of Integer Sequences) in several sections. The growth of the book itself is revealing: the first edition contained about 161 pages, the second about 287, while the third reached 438 pages—reflecting both progress on old problems and the continual emergence of new ones.
What makes the book particularly valuable is Guy's philosophy that mathematics advances as much through asking good questions as through answering them. It is therefore especially useful for advanced students, teachers, problem solvers, and researchers searching for ideas that might develop into projects or papers. It is not a conventional textbook and many problems require substantial background to attack seriously, but browsing it can be fascinating even when the mathematics eventually becomes difficult. Reviewers have specifically praised it as a source from which younger mathematicians can discover genuine research problems.
Key takeaways
Springer — Unsolved Problems in Number Theory
Goodreads — Unsolved Problems in Number Theory
Author: Richard K. Guy
Publisher: Springer
Publication: 3rd edition, 2004
Series:Problem Books in Mathematics
Richard K. Guy’s Unsolved Problems in Number Theory is a classic reference devoted not to teaching number theory systematically, but to showing where our knowledge of integers reaches its limits. Guy assembles a remarkably large collection of open questions, conjectures, partial results, computational observations, and references. Many problems are extremely easy to state yet extraordinarily difficult to solve—one of the characteristic attractions of number theory. The book is organized around six broad areas: prime numbers, divisibility, additive number theory, Diophantine equations, integer sequences, and miscellaneous problems.
The prime-number sections include questions concerning prime-producing formulas, Mersenne and Fermat numbers, prime gaps, arithmetic progressions of primes, pseudoprimes, and patterns among consecutive primes. Other chapters explore divisibility properties, representations of integers as sums, Diophantine equations, and unusual integer sequences. Rather than simply presenting a catalogue of puzzles, Guy explains what was known about each question, records related results and conjectures, and points readers toward the mathematical literature. Consequently, the book can function as a map of research problems: a reader can start with an elementary-looking question and quickly discover the deeper mathematics surrounding it.
The third edition substantially expanded the earlier versions, adding problems involving symmetric and asymmetric primes, sums of higher powers, Diophantine $m$-tuples, Conway's RATS and palindromes, together with references to the OEIS (Online Encyclopedia of Integer Sequences) in several sections. The growth of the book itself is revealing: the first edition contained about 161 pages, the second about 287, while the third reached 438 pages—reflecting both progress on old problems and the continual emergence of new ones.
What makes the book particularly valuable is Guy's philosophy that mathematics advances as much through asking good questions as through answering them. It is therefore especially useful for advanced students, teachers, problem solvers, and researchers searching for ideas that might develop into projects or papers. It is not a conventional textbook and many problems require substantial background to attack seriously, but browsing it can be fascinating even when the mathematics eventually becomes difficult. Reviewers have specifically praised it as a source from which younger mathematicians can discover genuine research problems.
Key takeaways
- Number theory remains full of deceptively simple open questions: elementary statements can conceal very deep mathematics.
- The book is a research catalogue rather than a standard textbook: problems are accompanied by known results, context, and extensive references.
- It covers a remarkably broad spectrum, from primes and divisibility to additive problems, Diophantine equations and integer sequences.
- It is particularly good for discovering research directions: a student can use Guy's references to move from recreational experimentation toward serious mathematical literature.
Springer — Unsolved Problems in Number Theory
Goodreads — Unsolved Problems in Number Theory
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│ KONSTANTINOS MICHAILIDIS │
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