Overcoming Challenges of Self-Studying Number Theory

Introduction

During my summer break I spent several hours each day self-studying mathematics—primarily number theory—even though I had no prior experience in theoretical math. The struggle of learning is not unique to mathematics; I faced many similar challenges during my first year of computer engineering (which I had just completed when I wrote this article).

There are many difficulties we must overcome to succeed. In this article I describe the challenges I faced while self-learning (many of which will be familiar to others from different academic backgrounds), how I overcame most of them, and one challenge I have not yet solved.

Challenge #1: Self-Doubt

The first—and by far most destructive—obstacle was self-doubt. Whenever I encountered difficult material I felt like giving up. I had the nagging thought that I wasn’t smart enough, that I would fail, and that I should stop wasting my time.

What helped

I realized much of the doubt came from comparing myself to people who seemed to breeze through the material, many of whom had more experience. To reduce its impact I stopped making those comparisons and focused on persistence. I kept working through problems; when I eventually solved them I gained confidence. Although self-doubt may never fully disappear, its effect lessens with time and experience if you persist.

Challenge #2: Loss of Interest

Losing interest in material was a direct consequence of doubt. When I struggled with topics or felt I wasn’t progressing fast enough, I became frustrated and started rationalizing why the material wasn’t worth learning. For example, I told myself that number theory had no real-life application and that I should focus on more important things.

What helped

After addressing my self-doubt I began to regain interest. I tracked my progress and began looking forward to learning number theory and the mathematical maturity it would give me—useful across science and theoretical computer science. When interest wanes, give yourself a concrete outcome to look forward to from the topic you are studying.

Challenge #3: Time Management and Unrealistic Goals

I initially set an aggressive quota of pages to complete per day and pictured finishing many books by the end of summer. That forced daily target made me less productive. If I hit a difficult proof I didn’t understand, I would skip it to meet the quota—only to later find that understanding the skipped proof was critical.

That cycle forced me either to keep skipping material or to stop and relearn prior sections. I realized my goals were unrealistic, so I committed to spending the time needed to truly understand concepts rather than racing through pages.

What helped

Adjust your goals to match your current skills. I accepted that I had never written a proof and lacked formal logic background, so I paused number theory and studied more basic topics (proof techniques, logic, sets) before returning. Be satisfied with realistic progress as long as you give full effort.

Challenge #4: Understanding Difficult Problems

I often could not understand complex problems because I was missing prior knowledge. When something confused me I identified the gap and returned to earlier lessons to relearn the material. Many seemingly unrelated or long proofs became manageable once I recognized the hidden subproblems and applied previous results.

What helped

The key is to break a giant problem into smaller subproblems and use prior solutions as building blocks. This approach turns intimidating problems into ones of reasonable difficulty, provided you have a firm grasp of the prerequisites.

Challenge #5: Lack of Feedback

The last challenge I still struggle with is the lack of timely feedback. Although online resources exist, prompt, accurate feedback on proofs is rare. There is no math “debugger”: when your final answer is wrong you may not know whether the error was a simple calculation mistake or a flawed logical approach. That ambiguity is frustrating, especially when time is limited.

What helps (so far)

I use the internet, forums, and study partners when possible, but the delay or lack of definitive feedback remains an unresolved pain point. Structured peer review, study groups, or mentors can help reduce this friction when available.

Conclusion

Overcoming these challenges is a vital part of learning. In my experience, the most effective strategies were:

• Persist: don’t give up when material is hard.
• Stop unhealthy comparisons: focus on your own progress.
• Have something to look forward to from each topic (a concrete outcome).
• Set realistic goals that match your current skills and revise them as you improve.
• Know when to retreat and build prerequisite knowledge before continuing.
• Break hard problems into subproblems and reuse prior results.