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• Users: crd
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1. ### How to Proof?

assume p is a factor of ab. then p|ab. since p is prime, euclid's lemma says p|a and/or p|b, thus p is a factor of a and/or b.
2. ### Every natural number is sum of powers of 2

http://en.wikipedia.org/wiki/Ancient_Egyptian_multiplication
3. ### Every natural number is sum of powers of 2

Maybe I am missing something, but if we have shown that 1 is a power of 2, and for some natural number k=>1, we are assuming k is a sum of powers of 2, wouldn't k+1 necessarily be of the correct form since we are adding a power of two to a finite sum of powers of two we are left with a finite...
4. ### Lattice Points

Why not try looking at a circle of radius one centered at the origin, counting the points of interest there? Then look at a circle of radius 2 centered the origin, and count those points. Then a circle with radius 3, a circle with radius 4, radius 5, ..., radius n, and maybe you will be able to...
5. ### Lattice Points

http://mathworld.wolfram.com/CircleLatticePoints.html
6. ### Triangular matrices

It is stated in almost every linear algebra text i could find that the inverse of a triangular matrix is also triangular, but no proofs accompanied such statements. I am convinced that it is the truth, but I have not been able to write anything down that I am satisfied with that doesn't rely...
7. ### Lattice Points

you could draw a quadrant of a circle of radius 6 and check the number of points there and multiply that number by four, being careful not to double count points that lie on the axis, as for a closed form, i would be surprised if one did not exist... Side note Wolfram indeed does have a...
8. ### Is convolution a linear operator?

Maybe our definitions of convolutions are different, but the definition I have learned for the convolution of two functions is defined by an integral, i.e. $f*g(t)=\int_{0}^{t}f(u)g(t-u)du.$ Letting $v=t-u,$ then $dv=-du$($t$ is constant with respect to $u$). We also need to change our limits...