What do atoms, DNA, and prime numbers have in common? This was the question that launched the LUMS Math Circle session on April 23, 2026. Students discovered that just as atoms are the building blocks of matter and DNA carries the code of life, prime numbers are the building blocks of all whole numbers.

Led by Dr. Imran Anwar, the session transformed primes from a familiar classroom topic into a fascinating world of patterns, puzzles, and unsolved mysteries.

 Session Structure and Content

The session began with a discussion on why prime numbers matter, followed by one of nature’s most remarkable examples: the 13-year and 17-year cicadas. Students learned how these insects emerge after prime-numbered intervals, a strategy believed to help them avoid predators. This surprising connection between mathematics and nature immediately captured their attention.

To warm up, participants tackled the NRICH XAVI T-shirt problem, searching for patterns in multiples and discovering that numbers often hide structure beneath the surface.

The main exploration centered on the Sieve of Eratosthenes. Using the first 100 numbers arranged in ten columns, students crossed out composite numbers and watched the prime numbers gradually emerge. Along the way, they observed prime gaps and began asking deeper questions: Why are some gaps small while others are larger? Where do twin primes appear?

The investigation took an exciting turn when the same numbers were arranged into six columns. Students noticed that every prime greater than three appeared in only two columns and, through their own observations, discovered the famous 6k−1 and 6k+1 pattern. Extending the activity to a thirty-column sieve revealed even richer modular patterns and showed how changing perspective can uncover hidden mathematical structure.

The session concluded with Conway’s Prime Climbing Problem. Starting with numbers such as 20, 60, and 225, students repeatedly transformed numbers through their prime factorizations and watched them grow in unexpected ways. They also learned that although John Conway believed every climb eventually reaches a prime, a massive counterexample, 13,532,385,396,179, was discovered in 2017, adding an element of surprise to the story.

Learning Outcomes

By the end of the session, students:

• Applied the Sieve of Eratosthenes to identify prime numbers.

• Explored prime gaps, twin primes, and numerical patterns.

• Discovered why all primes greater than 3 must be of the form 6k−1 or 6k+1.

• Investigated modular arithmetic through different sieve arrangements.

• Experienced mathematical experimentation through Conway’s Prime Climbing Problem.

• Developed a deeper appreciation for the beauty, structure, and mystery of prime numbers.

Conclusion

From cicadas and prime gaps to sieves and Conway’s climbing problem, the session showed students that prime numbers are far more than a list of special integers. Through hands-on activities and guided discovery, participants experienced mathematics as a living subject—one filled with patterns, surprises, and questions that continue to intrigue mathematicians today.

Acknowledgments

The LUMS Math Circle team gratefully acknowledges Dr. Imran Anwar for conducting this engaging and thought-provoking session. Special thanks are extended to the Mathematics Department staff for their organizational support and to all participating students for their enthusiasm, curiosity, and active involvement throughout the session.

Certificates

At the conclusion of the session, certificates of participation were distributed among the students in recognition of their engagement in the activities.

Here are some highlights from the event: