Number theory for self-study students: books and computer languages

Benjamin Hutz has a recent book that could be appropriate: An Experimental Introduction to Number Theory.

This book presents material suitable for an undergraduate course in elementary number theory from a computational perspective. It seeks to not only introduce students to the standard topics in elementary number theory, such as prime factorization and modular arithmetic, but also to develop their ability to formulate and test precise conjectures from experimental data. Each topic is motivated by a question to be answered, followed by some experimental data, and, finally, the statement and proof of a theorem. There are numerous opportunities throughout the chapters and exercises for the students to engage in (guided) open-ended exploration. At the end of a course using this book, the students will understand how mathematics is developed from asking questions to gathering data to formulating and proving theorems.

Regarding languages, in the preface he mentions the following:

The book is not tied into any one computer algebra system, and there are several freely available. The systems SageMath and PARI/GP are two excellent freely available systems. Similarly, Mathematica, Maple, or other commercial software could also be used. Because there are many excellent tutorials for each of these systems freely available, they will not be presented in this book.

Seconding/complementing other answers: Python (and/or Python as a part of Sage) has a command-line interface (on Linux/Unix and on Mac OS) that does allow defining variables, pre-loading files that set things up, and so on. Python (and, thus, Sage) has built-in large integers that are easier to use than C++ large integers (in my opinion). And freely available.

EDIT: ... and (I forgot to mention) it is quite easy to use the built-in graphing and graphical capabilities of Sage (especially in the "notebook" mode).

For beginning number theory, Art of Problem Solving has an online course. The textbook used with it, Introduction to Number Theory by Matthew Crawford, can be used alone for self-study. (I have not used the course. I have used the textbook.) I tutored a very advanced 9-year-old using this book, and enjoyed it.

I see they also have an intermediate number theory course.

I would recommend Python combined with SageMath, as already recommended by Joseph O'Rourke, or rather SageMath and Python comes naturally.

Python is a modern, and widely used, interpreted language (no compilation needed) it supports big integers via the bignum type. (But using SageMath I think this is tangential, I mention it for completeness mainly.)

SageMath is a free CAS whose user language is essentially Python. SageMath has lots of number theory functionality; its founder William A. Stein is a number theorist.

He also wrote a nice book "Elementary Number Theory: Primes, Congruences, and Secrets" that uses this software; you can check it out on his site.

As you mentioned PARI/gp you might be interested to know that PARI is an integral part of SageMath.

What computer languages might one recommend for, say, investigations in number theory?

I find Mathematica ideal, e.g.:

• "Mod sequences that seem to become constant; and the number 316"
• "Does 53 diverge to infinity in this Collatz-like sequence?"

But: (a) there is a huge start-up learning curve, and (b) Mathematica is not free. Because of the latter, I recommend Sage / SageMath:

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