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qedetc
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(nevermind, answered my own question after spending the time to type this up!)
Hi,
I was flipping through Hilbert's Geometry and the Imagination, and in it, he includes a proof of Leibniz' series ( pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... ) which is carried out by estimating the area of a circle at the origin using unit squares. I have a simple question about the appearance of a single " - 1" in the proof, but unfortunately need to describe half of the proof to ask about it. I assume I'm overlooking something extremely easy, but I just haven't been able to see why it's there.
The proof utilizes a theorem regarding the number of ways to express an integer n as the sum of squares of two integers: that this number is 4 times the quantity ( number of factors of n which are congruent to 1 mod 4 minus the number of factors of n which are congruent to 3 mod 4). In symbols,
S_n = | {(a,b) : a^2 + b^2 = n } | = 4 * (|{ d : d divides n and d = 1 mod 4 }| - |{ d : d divides n and d = 3 mod 4 }|)
where S_n is the number of ways to express n as the sum of two squares, and || is for the size of the set.
The main idea is to use this theorem to determine the number of unit squares whose bottom left corner is contained inside a circle of radius r; we'll call this number f(r). Now, Hilbert says that according to the above theorem, we can get (1/4)(f(r) - 1) by adding up the differences between the number of factors of the form 4k+1 and the number of factors of the form 4k+3 for all of the n <= r^2.
My question is regarding this "-1" in the "(1/4)(f(r) - 1)". Since you have a point inside/on the circle for each pair of integers (x,y) satisfying x^2 + y^2 <= r^2, it seems that f(r) should be the sum of the S_n for each integer n less than or equal to r^2. Then f(r) - 1 would be the sum over all of the S_n, minus 1. Further, (1/4)(f(r) - 1) would be the sum over all of the S_n, minus 1, all divided by 4; that is, -(1/4) + the sum of the differences between the number of factors of the form 4k+1 and the number of factors of the form 4k+3 for each n. So, it looks to me like that -1 doesn't fit (because it causes f(r) to differ from the sum Hilbert claims by -(1/4). I don't see where it came from, but I imagine I'm overlooking something incredibly simple.Thanksedit: Well, after typing this out, I reread the section again and noticed I had been completely ignoring that he starts discussing only the positive n's. Subtracting the 1 then takes care of the (0,0) solution.
Hi,
I was flipping through Hilbert's Geometry and the Imagination, and in it, he includes a proof of Leibniz' series ( pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... ) which is carried out by estimating the area of a circle at the origin using unit squares. I have a simple question about the appearance of a single " - 1" in the proof, but unfortunately need to describe half of the proof to ask about it. I assume I'm overlooking something extremely easy, but I just haven't been able to see why it's there.
The proof utilizes a theorem regarding the number of ways to express an integer n as the sum of squares of two integers: that this number is 4 times the quantity ( number of factors of n which are congruent to 1 mod 4 minus the number of factors of n which are congruent to 3 mod 4). In symbols,
S_n = | {(a,b) : a^2 + b^2 = n } | = 4 * (|{ d : d divides n and d = 1 mod 4 }| - |{ d : d divides n and d = 3 mod 4 }|)
where S_n is the number of ways to express n as the sum of two squares, and || is for the size of the set.
The main idea is to use this theorem to determine the number of unit squares whose bottom left corner is contained inside a circle of radius r; we'll call this number f(r). Now, Hilbert says that according to the above theorem, we can get (1/4)(f(r) - 1) by adding up the differences between the number of factors of the form 4k+1 and the number of factors of the form 4k+3 for all of the n <= r^2.
My question is regarding this "-1" in the "(1/4)(f(r) - 1)". Since you have a point inside/on the circle for each pair of integers (x,y) satisfying x^2 + y^2 <= r^2, it seems that f(r) should be the sum of the S_n for each integer n less than or equal to r^2. Then f(r) - 1 would be the sum over all of the S_n, minus 1. Further, (1/4)(f(r) - 1) would be the sum over all of the S_n, minus 1, all divided by 4; that is, -(1/4) + the sum of the differences between the number of factors of the form 4k+1 and the number of factors of the form 4k+3 for each n. So, it looks to me like that -1 doesn't fit (because it causes f(r) to differ from the sum Hilbert claims by -(1/4). I don't see where it came from, but I imagine I'm overlooking something incredibly simple.Thanksedit: Well, after typing this out, I reread the section again and noticed I had been completely ignoring that he starts discussing only the positive n's. Subtracting the 1 then takes care of the (0,0) solution.
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