March 4, 2025 | Janaki Pande, Asian College of Journalism
UK Anandavardhanan, a Professor of Mathematics at IIT Bombay, presents a talk on number theory to the general public at Science at the Sabha 2025. (Photo: IMSc Media)
The building blocks
Anyone who has sat through maths classes in school will be familiar with natural numbers, rational numbers, and right-angled triangles. Though seemingly unrelated, these basic concepts are all one needs to understand one of number theory’s ancient unsolved problems: the Congruent Number Problem.
For those whose school days are a distant blur, a natural number is any whole number except \(0\) (for example, \(1\), \(2\), \(3\) and so on to infinity). A positive rational number is a fraction made up of two natural numbers (for example, \(\frac{1}{2}\), \(\frac{5}{7}\), \(\frac{19}{25}\)). A right-angled triangle can be thought of as one half of a rectangle sliced diagonally.
Under the scrutiny of curious mathematicians, interesting patterns and questions emerge from these basic concepts. These problems are sometimes easy to state, yet their answers are the subject of years of research. The Congruent Number Problem is one such puzzle that was first mentioned in an equivalent form in a 10th century Persian manuscript. Since then it has surfaced in the notebooks of mathematical legends like Fibonacci and Fermat.
On a warm Sunday afternoon at the Music Academy in Chennai, the Congruent Number Problem is evoked again before a gathering of the general public including young children. In a hall that often resonates with classical melodies, UK Anandavardhanan, a Professor in mathematics at the Indian Institute of Technology (IIT) Bombay begins his lecture, ‘A Gentle Invitation into Number Theory’. It is a part of the eighth edition of the Science at the Sabha, a public outreach event organised by the Institute of Mathematical Sciences (IMSc), Chennai.
The Congruent Number Problem
“We have all heard of right-angled triangles,” says Anandavardhanan. “Can someone tell me what the Pythagorean theorem is?” A young boy in the crowd answers that it relates to a right-angled triangle, where the squares of the two shorter sides add up to the square of the longer side (for example, \(3^2 + 4^2 = 5^2\)). Anandavardhanan elaborates that the area of such a triangle can be calculated by multiplying its base by height and dividing this by two.
The building blocks of the Congruent Number Problem are right-angled triangles, natural numbers and rational numbers. In the image above, \(n\) is a congruent number if \(x\), \(y\) and \(z\) are rational numbers.
Coming to the crux of the problem, Anandavardhanan explains that a natural number is called a ‘congruent number’ if it can be expressed as an area of a triangle whose sides are rational numbers. In the figure above, \(n\) is a congruent number if \(x\), \(y\) and \(z\) are rational numbers.
While the concept of a congruent number might have been easy to grasp, the challenge lies in proving whether any given number is congruent or not.
Observations and advances
“Maths is also an experimental science, it’s all about pen and paper and deep thinking,” continues Anandavardhanan, as the first 50 congruent numbers appear on the screen. To mathematicians, these numbers are raw material for making observations, finding patterns and coming up with conjectures.
The first 50 congruent numbers. (Source: A003273 - OEIS shared under a CC by 4.0 license)
Transitioning into the second phase of the lecture, he gives his audience a glimpse into this process of mathematics research.
\(6\) is a congruent number because it is the area of a triangle with sides as the rational numbers \(3\), \(4\), and \(5\) (\(3^2 + 4^2 = 9 + 16 = 25 = 5^2\) and \(\frac{3 \times 4}{2}= 6\)). \(5\) is also a congruent number–it is the area of a right-angled triangle with sides as the fractions \(\frac{9}{6}\), \(\frac{40}{6}\), \(\frac{41}{6}\), which are also rational numbers. This makes us wonder if some numbers, like six, are more “easily congruent” than others.
The search for congruent numbers is not as easy as it may seem. Anandavardhanan points out that proving that \(1\), \(2\) and \(3\) are not congruent took centuries of research. After the initial mention of congruent numbers in Persian manuscripts, the Italian mathematician Fibonacci claimed that \(1\) and no other squares could be congruent numbers. This was proved by Fermat four centuries later. Fermat's method also proves that \(2\) is not congruent. It took many more decades of research to prove that \(3\) is not a congruent number.
Another observation is that if we scale a triangle associated with a congruent number by a natural number, it is possible to generate more congruent numbers. Scaling the sides of a triangle by a number increases the triangle’s area by the square of that number. For example, if we take a triangle (say, with sides \(3\), \(4\), \(5\), and area \(6\)) and double its sides (\(6\), \(8\), \(10\)), its area increases four times (\(24\)). Therefore, if the area of the initial triangle is a congruent number, the area of a triangle scaled by a natural number is the previous area multiplied by a perfect square, which is also a congruent number.
This narrows down the search for congruent numbers to ‘square-free numbers’, numbers whose prime factors do not repeat. For example, \(24\) can be written as \(2 \times 2 \times 2 \times 2 \times 3\), which is not a square-free number because \(2\) repeats itself. On the other hand, \(6\), which is \(2 \times 3\), is a square-free number. If a square-free number is proved to be congruent, then multiples of that number by a perfect square will also be congruent.
Over the years, the Congruent Number Problem has seen many developments often separated by decades of research. At present, it finds its way into modern mathematics through the world of elliptic curves and modular forms. The most recent development related to elliptic curves is a conjecture by Birch and Swinnerton-Dyer, which was categorised as a Millenium Prize Problem in 2000. It is among the seven mathematical problems selected by the Clay Mathematics Institute, for a $1 million award. Any progress towards settling the Birch and Swinnerton-Dyer conjecture in particular advances our understanding of the Congruent Number Problem.
The chase continues, for an elegant solution that will be able to determine whether a given natural number \(n\) is congruent or not.
The thrill of the chase
For many adults, maths remains confined to the classroom, the exam, the powdery remnants of chalk that float to the ground after blackboards are dusted off. “What is taught as maths in schools is not really what people like him do,” says Rahul Siddhartan, a faculty member at IMSc, about Anandavardhanan’s work.
Once you are out of the competitive system at school, and into a life of research, there is no hurry, Anandavardhanan explains. “You can take your time, meditate about ideas, and rethink things slowly,” he adds. To him, mathematics is about adding rigour to approaching questions. It is about looking at proofs, observing what works, and the excitement of possibly applying it to another seemingly disconnected problem.
Maths is also an experimental science, it’s all about pen and paper and deep thinking,” says Anandavardhanan. (Photo: IMSc Media)
Even among the academics in the audience, the Congruent Number Problem may have been new to many. In the scientific landscape, every subject and sub-field leads to profound but often isolated depths of research. Anuran Pal, a PhD student at IMSc, says, “Once you are past the barriers of studies and exams, there is a lot of freedom.” Scientists follow rigorous methods of inquiry and there are no forbidden questions or limits to their curiosity.
Researchers exist in a space where questions abound and the chase for answers never ceases. Sometimes, in events like these, they step out of their silos, to share, however briefly and simply, the problems that consume their days. Yet, as with the Congruent Number Problem, even if their research question seems easy to convey and grasp, the answers remain elusive, awaiting future exploration and breakthroughs.
Edited by Bharti Dharapuram
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Anandavardhanan's talk is available to watch online on the Matscience YouTube channel.
Janaki Pande can be contacted at janakipan02 [at] gmail [dot] com.
