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Jan
22
comment How to prove that all odd powers of two add one are multiples of three
(Just to fill in the last details: Let $k \in \mathbb{Z}$ be arbitrary. Among the three consecutive integers $2^{k} - 1, 2^{k}, 2^{k} + 1$ must be exactly one divisible by three; since $2^{k}$ is not divisible by three, it must be one of the other two, hence their product is divisible by three. This product is $(2^{k} - 1)(2^{k} + 1) = 2^{2k} - 1$; note that multiplying this number by $2$ and then adding $3$ does not affect its divisibility by three, whence $2(2^{2k} - 1) + 3 = 2^{2k+1} + 1$ is divisible by $3$ as desired. QED)
Jan
21
revised Defining vertical tangent lines
added 124 characters in body
Jan
20
awarded  Nice Answer
Jan
20
comment How to prove that all odd powers of two add one are multiples of three
@Glen_b Just for fun, I wrote out an answer using this sort of observation!
Jan
20
answered How to prove that all odd powers of two add one are multiples of three
Jan
10
answered A question about invertible elements and zero-divisors in a ring
Jan
6
comment When the product of dice rolls yields a square
In Class 3 part (3) what does "the remaining roll" refer to?
Jan
5
comment A finite set has no limit points
In R^n define m to be the minimum among all pairwise distances between points; the existence of m relies on your finiteness assumption. Then an open ball of radius m around any point p in your set contains only p, which shows p is not a limit point. So there are no limit points in the set.
Jan
4
accepted When the product of dice rolls yields a square
Jan
4
revised When the product of dice rolls yields a square
Linked to the WA diagonalization and its (0,0,0,2,2,2,4,6) diagonal
Jan
3
awarded  Popular Question
Jan
3
comment When the product of dice rolls yields a square
Ah yes, you are right; I added too hastily (and lost the $3$ coefficient of $e^{2x}$ in the process!). Quite separately: Have you ever seen a problem of this nature solved "directly" (I'm not quite sure what this would mean) using hyperbolic functions? What I have in mind is, as you solved the problem using generating functions, sinh and cosh arose; is there a (non-contrived) way to solve this problem that avoids the explicit use of generating functions and re-formulates it into sinh and cosh from the outset?
Jan
2
comment Is there a branch of Mathematics which connects Calculus and Discrete Math / Number Theory?
I would say that statistics and probability do a pretty good job of taking discrete data, assuming some underlying continuous feature, and extracting results via Calculus or its higher analogues (e.g. Measure Theory). There are other areas of math that involve something like connections between the zeta function (number theory) and counting points (discrete mathematics) in finite fields (algebra and algebraic geometry). (I have in mind the Weil Conjectures as a concrete example, but I think the phenomenon of moving between the discrete and continuous is rather common.)
Jan
2
awarded  Nice Question
Dec
30
revised When the product of dice rolls yields a square
Based on the most recent answer, I have added the generating-functions tag.
Dec
30
comment When the product of dice rolls yields a square
Excellent. Do you have a recommended reference for problem solving with generating functions? (I would be especially interested in seeing your Steps 4 and 5 applied similarly to other problems...)
Dec
29
comment When the product of dice rolls yields a square
The diagonalization approach is very nice; thank you!
Dec
28
asked When the product of dice rolls yields a square
Dec
28
answered What's the smallest number that we can multiply with a given one to get the result only zeros and ones?
Dec
27
comment Divisibility by 37 .
If you should face down a problem about $37$ again: The observation that $3 \times 37 = 111$ is usually relevant (as it was/is here).