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F. R. David was born on 1947-01-01.

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F. R. David was born on 1947-01-01.

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Suppose N is a perfect number. Then N cannot be a square number and so N has an even number of factors.Suppose the factors are f(1) =1, f(2), f(3), ... , f(k-1), f(k)=N.

Furthermore f(r) * f(k+1-r) = N for r = 1, 2, ... k so that f(r) = N/f(k+1-r)

which implies that 1/f(r) = f(k+1-r)/N


Then 1/f(1) + 1/(f(2) + ... + 1/f(k)

= f(k)/N + f(k-1)/N + ... + f(1)/N

= [f(k) + f(k-1) + ... + f(1)] / N

= 2N/N since, by definition, [f(k) + f(k-1) + ... + f(1)] = 2N

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David R. Brower was born on July 1, 1912.

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David R. Brower was born on July 1, 1912.

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f(x) = x^{2} is a continuous function on the set R of real numbers, and (-1, 1) is an open set in R, but f(-1, 1) = [0, 1), and [0, 1) is not an open set in R. So, f is not an open function on R.

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