Lucas' theorem: elementary number theory, and useful in…
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Lucas' theorem: elementary number theory, and useful in modern research!
Pascal's triangle, with entries colored according to parity https://preview.redd.it/wgnr7fq4rkgh1.png?width=1218&format=png&auto=webp&s=587db0b1678bafbbee663eb3863fab5056eeb648
If you take *Pascal's triangle*, and color each entry according to whether it is even or odd, you get a funny pattern, which resembles Sierpinski's triangle. To understand this, it's helpful to know *Lucas' theorem*, which tells you when an entry in Pascal's triangle will be even or odd. If you've never seen it, you might enjoy the article https://hidden-phenomena.com/articles/lucas that we just wrote about it!
Lucas' theorem is a great result, which even tells you about how to compute (n choose k) modulo p. It is a wonderful piece of elementary number theory, and suitable as a fun but challenging exercise for the end of an elementary number theory course. Recently, one of us had to invoke Lucas' theorem in a modern math research paper https://arxiv.org/abs/2604.20054 about some relatively fancy arithmetic geometry! We thought this was a good example of how small results from introductory courses can be helpful in your research career in completely unexpected ways! The article itself isn't about the paper (which isn't very elementary), but Lucas' theorem is still helpful, and will hopefully come in handy.
#science
Pascal's triangle, with entries colored according to parity https://preview.redd.it/wgnr7fq4rkgh1.png?width=1218&format=png&auto=webp&s=587db0b1678bafbbee663eb3863fab5056eeb648
If you take *Pascal's triangle*, and color each entry according to whether it is even or odd, you get a funny pattern, which resembles Sierpinski's triangle. To understand this, it's helpful to know *Lucas' theorem*, which tells you when an entry in Pascal's triangle will be even or odd. If you've never seen it, you might enjoy the article https://hidden-phenomena.com/articles/lucas that we just wrote about it!
Lucas' theorem is a great result, which even tells you about how to compute (n choose k) modulo p. It is a wonderful piece of elementary number theory, and suitable as a fun but challenging exercise for the end of an elementary number theory course. Recently, one of us had to invoke Lucas' theorem in a modern math research paper https://arxiv.org/abs/2604.20054 about some relatively fancy arithmetic geometry! We thought this was a good example of how small results from introductory courses can be helpful in your research career in completely unexpected ways! The article itself isn't about the paper (which isn't very elementary), but Lucas' theorem is still helpful, and will hopefully come in handy.
#science
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