Imagine seeing the following calculation: The correct result is and so the first calculation is off by 25 orders of magnitude.
But there’s a variation on the calculation above that is correct! A theorem by Édouard Lucas from 1872 that says for p prime and for any nonnegative integers m and n, So while the initial calculation was grossly wrong as stated, it is perfectly correct mod 19.
If you divide 487343696971437395556698010 by 19 you’ll get a remainder of 10.
A stronger versions of Lucas’ theorem [1] says that if p is at least 5, then you can replace mod p with mod p³.
This is a stronger conclusion because it says not only is the difference between the left and right side of the congruence divisible by p, it’s also divisible by p² and p³.
In our example, not only is the remainder 10 when 487343696971437395556698010 is divided by 19, the remainder is also 10 when dividing by 19² = 361 and 19³ = 6859.
More on binomial coefficients Binomial coefficient trick Maximum gap between binomial coefficients Lebesgue’s proof of Weierstrass’ theorem [1] V.
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On the divisibility of the difference between two binomial coefficients, Den 11te Skandinaviske Matematikerkongress, Trondheim, 1949, 42–54.
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