Mathematicians Settle a Long-Standing Question About Even and Odd in Number Theory
Some of the deepest questions in mathematics are about even and odd. Not the even and odd of a times table, but a hidden sign — a plus one or a minus one — buried inside an equation, which mathematicians believe predicts whether the number of solutions to that equation is even or odd. A new paper proves that prediction holds for a broad new class of cases.
The paper, "p-parity conjecture for Panchishkin Galois representations", was posted to the preprint server arXiv on 5 November 2026 by Subham Garai, Somnath Jha and Sudhanshu Shekhar, mathematicians at the Indian Institute of Technology Kanpur. It has not yet been peer-reviewed, which is normal for a fresh preprint — the checking by other mathematicians is the next step.
What is a parity conjecture?
Imagine an equation with whole-number solutions. Counting those solutions exactly can be enormously hard. But sometimes you do not need the exact count: you only need to know whether it is even or odd. That is parity.
Number theorists attach a number called a root number to certain equations and their associated objects. The root number is always either +1 or −1. The parity conjecture says the root number is a perfect fortune-teller: if the root number is +1, the count of solutions (measured in a precise way, through something called a Selmer group) is even; if it is −1, the count is odd.
The objects in this paper are called Galois representations — abstract structures that encode how the symmetries of number systems act. The team proved the "relative" version of the conjecture: for two such representations that look the same in a certain technical sense, the ratio of their root numbers equals the parity of the difference in the sizes of their Selmer groups. The result applies to a broad class — symplectic self-dual representations that are pure of weight −1 and satisfy a condition named after the mathematician Panchishkin — with no restriction on their dimension. It gives new cases of the conjecture for situations of rank greater than two, including twists of modular forms and families of abelian surfaces.
Why prove something so abstract?
Parity results are shortcuts. If a sign can tell you whether a count is even or odd, you can rule out whole families of answers without ever doing the full count — and a wrong sign would expose a mistake in the theory. Each new case where the conjecture is proved strengthens the bridge between two sides of number theory: the analytic side (the root number, coming from functions called L-functions) and the arithmetic side (the Selmer group, counting solutions). That bridge is at the heart of some of the biggest unsolved problems in mathematics, including the Birch and Swinnerton-Dyer conjecture, one of the Clay Millennium Prize Problems.
What this means for students
Parity is one of the first powerful ideas children meet in maths: odd plus odd is even, even times anything is even. This paper is that same idea, grown up — mathematicians using even and odd as a detective tool to check answers they cannot yet count. It is a reminder that the simple ideas from primary school never stop being useful; they just get applied to harder and harder puzzles.
If your child enjoys spotting patterns in numbers and wants to see how far a simple idea like even and odd can reach, a first lesson in maths can take them from parity puzzles to the questions mathematicians are still solving today. Book your first lesson
Sources:
- Garai, Jha and Shekhar, "p-parity conjecture for Panchishkin Galois representations", arXiv:2611.08316, submitted 5 November 2026: https://arxiv.org/abs/2611.08316
- arXiv News, "Mathematicians Prove a General Parity Relation for a Broad Class of Galois Representations", 8 November 2026: https://www.arxivnews.org/en/articles/8eab0b45-f974-4e71-bd61-753709a58e18
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