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Recreational Curiosities to Unsolved Conjectures: A Review of Manjul Bhargava’s “Patterns, in numbers and nature, inspired me to pursue mathematics”

By Rupert Li

Prime numbers must be a manmade construct, right? Few animals possess the mental faculties to count, much less be concerned about whether an integer has any nontrivial factors. But then why do North American cicadas follow either a 13 or 17-year life cycle? One of the most prominent hypotheses is that choosing prime numbers reduces risk of hybridization, i.e., the mixing of two broods of cicadas. With cicadas emerging every 13 and 17 years, they only appear in the same season once every 13*17=221 years. If we instead took two composite numbers, such as 12 and 18, we would find that broods hybridize once every 36 years, more than six times as frequently. Number theorist Prof. Manjul Bhargava of Princeton University, the first Fields medalist of Indian origin, sees this natural pattern and others as a source of inspiration to pursue mathematics, whose essential nature is discovering and explaining patterns. In his video “Patterns, in numbers and nature, inspired me to pursue mathematics,” Bhargava presents incontrovertible mathematical truths in nature as an illustration of how recreational curiosities can quickly lead to deep mathematics at the frontiers of human knowledge.

Math and numbers appear in all aspects of life, even in language. Sanskrit poetry, which Bhargava studied and deeply admires, consists of long, two-beat syllables and short, one-beat syllables. Thus, poets are interested in possible rhythmic patterns within a given number of beats. For example, how many patterns of short and long syllables span exactly eight beats? The answer turns out to be a Fibonacci number, as Sanskrit linguists had known long before Fibonacci’s time. In fact, the Fibonacci numbers repeatedly appear in nature as well: any natural spiral, such as a pinecone or a sunflower, will have a Fibonacci number of spirals—regardless of whether you look at it clockwise or counterclockwise! Understanding why patterns like these occur is fundamental to the pursuit of mathematics, providing reassurance that patterns are not mere coincidence and deepening our understanding of the underlying mechanisms of nature. Conversely, one’s diverse set of experiences, even those in seemingly unrelated areas, can provide valuable insight into mathematics.

Beyond observing patterns in the natural world around him, Bhargava to this day also enjoys finding curious patterns in numbers, a sentiment we can all relate to. One of his favorite numbers as a child was 142857. If you multiply this number by 2, you get 285714; multiplying by 3 yields 571428, and similarly multiplying by 4, 5, or 6 all yield some number obtained by reading 142857 from some starting point, wrapping around at the end (you may recognize 142857 as the repeating decimal representation of the fraction 1/7). Naturally, one may ask if there are other so-called cyclic numbers, to which the answer is yes: you can multiply the sixteen-digit number 0588235294117647 by 1 through 16 to obtain cyclic shifts of the same number. Surprisingly, whether there are only a finite set of cyclic numbers is still an unsolved problem in mathematics.

Some may argue that there is a wide schism between recreational math problems like these and modern-day math research, that nowadays math has outgrown its avocational, natural pattern-driven roots to become a lofty world disconnected from reality. However, Bhargava argues that mathematical curiosities that we enjoyed as children are still closely connected to state-of-the-art math research. For instance, a major unsolved problem in number theory is Artin’s conjecture. If proven, Artin’s conjecture would imply that there are infinitely many cyclic numbers, so in many ways the patterns Bhargava observed at a young age continue to inspire the mathematics he studies decades later. Diving into why patterns such as Fibonacci or cyclic numbers occur offers powerful insights into even deeper mathematics. And ultimately, sequences and structures in nature are denizens of mathematical truth. So, I encourage you to always be open to observing new and interesting patterns, and to never stop wondering: why?