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1 vote

Hermite's identity for sum of floor function

Assume that a function $f$ is given as a power series $$ f(x) = A_0 + A_1x + A_2x^2 + \ldots; $$ then $$ \frac{f(x)}{1-x} = A_0 + (A_0+A_1)x + (A_0 + A_1 + A_2)x^2 + \ldots $$ as can be verified ...
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1 vote

This seems blatantly false.

It does sound like you're hung up on the wording. I believe he's saying as many "primes" (including 1 in his definition) as one wishes up to the integer itself. To be specific, $$\exists \...
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  • 11
0 votes

This seems blatantly false.

It seems quite clear to me, once we accept that $1$ is a prime. We have $$5=3+2$$ $$5=3+1+1$$ $$5=1+1+1+2$$ $$5=1+1+1+1+1$$ So $5$ can be written as the sum of $2,3,4,$ or $5$ primes, until we get to $...
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7 votes

This seems blatantly false.

The very next sentence on the Wikipedia page you quoted states the answer: Goldbach was following the now-abandoned convention of considering 1 to be a prime number, so that a sum of units would ...
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  • 368
1 vote

Can you provide me historical examples of pure mathematics becoming "useful"?

I am not sure that whether this is a real application or not but this is one of reasons that I don't want to exit from mathematics: Question: Is that possible to monitor each point on Earth in real ...
0 votes

From a given point Α outside a line α how many lines parallels to the line α can you draw?

Assuming you are working in 2-dimensional Euclidean geometry the answer is no. This is in fact an axiom of Euclidean geometry, it is something we assume to be true a priori, not a property which we ...
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  • 2,266
0 votes

Equivalency between a "mixed modular equation" of Gauss and a later theorem of Ramanujan.

I just want to describe the arithmetic information yielded by Gauss's identities. As in Somos's and Paramanand Singh's answers, Gauss's identity translates into the equation: $$(3\phi^2(q^9)-\phi^2(q))...
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  • 830
2 votes

Why are the trig functions versine, haversine, exsecant, etc, rarely used in modern mathematics?

It’s true that those functions are rarely used today for the reasons set out above. But, they are still used. For example, there are many yachts cruising around the world and while most, if not all, ...
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  • 21
4 votes
Accepted

Did Newton and Leibniz use limits in their derivations of differential calculus?

Newton and Leibniz initially expressed the derivative $f'(x)$ as the ratio $\frac{df}{dx}$, where the rates of change $df$ and $dx$ are infinitesimal numbers. Analogously, the integral was defined as ...
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  • 1,535
1 vote

How to prove an identity of Gauss for the logarithm of Jacobi theta function?

I somehow missed that trick in my previous attempts but franz lemmermeyer's useful comment was a sort of guidance that made it clear to me how to resolve this question. Lets look at the seperate cases ...
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  • 830
1 vote

How to prove an identity of Gauss for the logarithm of Jacobi theta function?

This is not an answer and I've got to run. Take $n$ odd; then $$ \frac{2q^n - q^{2n}}{1-q^{2n}} = \frac{2q^n - 2q^{2n}}{1-q^{2n}} + \frac{q^{2n}}{1-q^{2n}} = 2 \frac{q^n(1-q^n)}{ 1-q^{2n}} + \...
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0 votes

Old Math Books Request

Two classic and influential ones on abstract algebra which are still missing in the other answers: Heinrich Weber: Lehrbuch der Algebra. Vol. 1 (1895), Vol. 2 (1895), Vol. 3 (1898) Bartel van der ...
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  • 20.6k
7 votes
Accepted

Origin of quote “In order to simplify…”

I'm going to deviate from my usual policy of never answering a question because I can't "really" post images in comments, and complex google searches may yield different results for ...
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2 votes

Are there cases where a flawed proof seems correct?

It does seem to happen on a regular basis that a physics-y argument is given that purports to prove the Riemann Hypothesis... but, also, at the same time would seem to prove some other things which we ...
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  • 47.5k
0 votes

Have there been any comprehensive studies on the language of mathematics?

The reason mathematics is difficult for the untrained to understand is obviously not because of the language but because of the logical reasoning involved. One must have a sufficiently good grasp of ...
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  • 54.1k
2 votes
Accepted

Are there cases where a flawed proof seems correct?

A number of results in analysis were proven then counterexamples were later found. For example Ampere proved in 1806 that any continuous function from $\mathbb{R}$ to itself is differentiable ...
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