09-15-2026, 12:49 AM
Solved and Unsolved Problems in Number Theory — Daniel Shanks
Shanks's central philosophy is that elementary questions can lead surprisingly far into genuine research mathematics. He starts with problems that can be understood using relatively accessible number theory, develops the known results, and then shows how they naturally generate new questions, conjectures, and unsolved problems. Rather than presenting number theory as a collection of finished theorems, the book emphasizes the process of mathematical discovery.
The three principal themes are:
What makes the book unusual
A particularly valuable feature is that Shanks doesn't stop when a problem has been "solved." He repeatedly asks what happens next? A theorem often becomes the starting point for another conjecture or computational investigation. The later material incorporates research carried out between 1962 and 1978, including results obtained with the assistance of computers.
So the book sits somewhere between a textbook, a collection of problems, and a historical account of mathematical research. It illustrates how computation can suggest patterns and conjectures, while rigorous mathematics is needed to establish them.
Mathematical level
Despite its research flavor, much of the starting material is elementary number theory. You do not need advanced abstract algebra to appreciate the basic problems. However, some of the developments become substantially deeper, making the book particularly interesting for someone who wants to progress from olympiad-style number theory toward research-level questions.
BOOK
Shanks's central philosophy is that elementary questions can lead surprisingly far into genuine research mathematics. He starts with problems that can be understood using relatively accessible number theory, develops the known results, and then shows how they naturally generate new questions, conjectures, and unsolved problems. Rather than presenting number theory as a collection of finished theorems, the book emphasizes the process of mathematical discovery.
The three principal themes are:
- Perfect numbers: numbers equal to the sum of their proper divisors. Their study leads to questions about the structure and existence of perfect numbers and their relationship with primes.
- Periodic decimals: the repeating structure of rational decimal expansions is connected to divisibility, modular arithmetic, and the multiplicative behavior of integers.
- Pythagorean numbers: integer solutions of equations such as $x^2+y^2=z^2$ lead into classical Diophantine problems and broader questions about representations of numbers.
What makes the book unusual
A particularly valuable feature is that Shanks doesn't stop when a problem has been "solved." He repeatedly asks what happens next? A theorem often becomes the starting point for another conjecture or computational investigation. The later material incorporates research carried out between 1962 and 1978, including results obtained with the assistance of computers.
So the book sits somewhere between a textbook, a collection of problems, and a historical account of mathematical research. It illustrates how computation can suggest patterns and conjectures, while rigorous mathematics is needed to establish them.
Mathematical level
Despite its research flavor, much of the starting material is elementary number theory. You do not need advanced abstract algebra to appreciate the basic problems. However, some of the developments become substantially deeper, making the book particularly interesting for someone who wants to progress from olympiad-style number theory toward research-level questions.
BOOK
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