[–] 0 points1 point  (0 children)

Are you asking us to break someone's RSA key? Your number is 200 digits long (665 bits), and looking at the RSA Factoring Challenge, a number roughly this same size (RSA-200 at 663 bits) took the equivalent of 75 years of computing time on a 2.2 GHz Opteron processor back in 2005. I don't think anyone's going to solve that here.

[–] 0 points1 point  (0 children)

Question 5.

You want to find the values of x for which f(x) < g(x), so replacing f(x) and g(x) by their expressions, we want to find the values of x for which x(x-1) < x. By rearranging, we get x2-2x < 0. The left hand side is a quadratic curve with roots at x=0 and x=2. If we check at x=1, we see that (1)2-2·1 = -1 is negative, and so we know that the curve x2-2x goes from positive to negative when crossing x=0 and then from negative to positive when crossing x=2. Hence the solution is that f(x) < g(x) whenever 0 < x < 2.

Question 16.

Because the function is defined by parts, we will analyse the range over two intervals. First, when x < 0, f(x) = 3·sin(x), and since the range of sin(x) is [-1,1], the range of f(x) is going to be [-3,3] when x < 0. However, when x ≥ 0, f(x) = sqrt(x), and the range of sqrt(x) is [0,∞). If we combine these two results, f(x) has a range of [-3,∞) over its entire domain.

Question 46.

First let's factor the numerator: x2-5·x+6 = (x-2)(x-3). Therefore, f(x) = (x-2)(x-3)/(x-2), which means that f(x) = x-3 whenever x is different from 2, and f(2) is undefined. Since f(x) is linear, it only has one x-intercept. Now the graph of f(x) and of x-3 look a lot alike, but you have to remember that f(2) is not defined, and so their graphs are distinct at x=2. Further, because of that difference, (2-3) = -1 is not in the range of f(x), though every other real number is.

Question 49.

We know that f(x) = sin(x) is an odd function, which means that f(-x) = -f(x). So since we want to know when f(-x) = f(x), this is the same as asking when -f(x) = f(x), i.e. when is f(x) = 0? This, we know to be the integer multiples of pi.

Question 50.

We have an intersection of the x-axis wherever we have a root of x(x2-2)(x2+x+1). This is a matter of factor the expression. Note that x2-2 = (x-sqrt(2))(x+sqrt(2)), and that x2+x+1 does not factor any further (that is because b2-4·a·c = 1-4 = -3 < 0). For that reason, we have that x(x2-2)(x2+x+1) = (x-0)(x-sqrt(2))(x+sqrt(2))(x2+x+1) = 0, which then clearly has 3 solutions: x = 0, x = -sqrt(2) and x = sqrt(2).

[–] 0 points1 point  (0 children)

I've seen Z_p used for the p-adic integer (for p prime), so I usually tend not to use this notation, but in most cases there is no confusion to be had between the two objects anyway.

[–] 0 points1 point  (0 children)

I'm not sure if this one tends too much towards the textbook kind, but From Fermat to Minkowski (original Von Fermat bis Minkowski) by Scharlau & Opolka gives a very nice historical introduction to number theory. The book is aimed at last-year undergrad student, so some knowledge from the reader is assumed (some field theory and ring theory).

[–] 1 point2 points  (0 children)

I'd also be interested in joining this group. I don't know any of the proposed platforms (besides reddit), but Slack doesn't seem to fit the "classroom" model based on its introductory video.

[–] 0 points1 point  (0 children)

Well I was assuming f was a function from R2 to R, and the question specifies that f(x,x) ∈ Z iff x ∈ Z, so the sines are to make sure that f(x,x) ∈ (0,1) for x ∈ R\Z. I might be assuming too many restrictions, though...

[–] 0 points1 point  (0 children)

The real numbers are usually considered to be the “∞-adics”, i.e. the completion of Q with respect to the “∞-adic” metric |·|_∞ = |·|. The reason for this convention arises from an analogy that is nicely explained here. The wiki page is also not bad.

The reason why Conway refers to those as the (−1)-adics is explained in the preface of the book. He writes:

This reminds me of the fact that some people always smile indulgently when I mention “the prime −1”, and continue to use what they presume to be the grown up name “∞”. But consider:

Every nonzero rational number is uniquely a product of powers of prime numbers p.

For distinct odd primes (p/q) and (q/p) differ just when p ≡ q ≡ −1 (mod 4).

There is an invariant called the p-signature whose definition involves summing p-parts of numbers.

If there are p-adically integral root vectors of norm k and kp, then p is in the spinor kernel.

Each of these statements includes the case p = −1, but none of them is even meaningful when we use the silly name “∞”. In the future, I shall smile indulgently back!

As for your follow-up question, I have no idea. Sorry.

[–] 2 points3 points  (0 children)

[–] 2 points3 points  (0 children)

If I'm correct, the "sum" is defined by

x + y + z = (xy)::(xz)::(xy+xz-y),

where :: denotes concatenation. Based on this, the solution is

8 + 3 + 9 = (8*3)::(8*9)::(8*3+8*9-3) = 24::72::(24+72-3) = 247293.

It's possible however for there to be other rules. In particular, for any number α, you can define polynomial in three variables such that the two triples (5,3,2) and (7,2,5) give the correct answer, and the last triple (8,3,9) give α.

In that sense, you could say that there are uncountably many solutions.

[–] 4 points5 points  (0 children)

I think you are right, and that there is a slight error. That being said, the proof is still essentially correct: if you look at the mean value theorem, it only requires for F to be continuous on the closed interval [a,b] and differentiable on the open interval (a,b), which, as you've mentioned, is given by assumption. Similarly, F is continuous on each closed [x_i-1, x_i] and differentiable on each open (x_i-1, x_i). Good catch!

Someone should probably fix it. I know Wikipedia's rules are sometimes very stringent, so I won't dare do it myself.

[–] 0 points1 point  (0 children)

First of all, you want to only consider the relevant resistances. You don't really care for the flow going from the brewery to the first bottling facility, since it does not vary depending on the location of the junction point (its value is always k·4/(0.8)4). The only flow that matters is the one from the junction to the second bottling facility.

If d is the distance from the brewery to J, then you can easily find the distance between J and the second bottling facility, and consequently you can also find a nice equation for the resistance in that pipe. Since it's the only resistance that matters, you can minimise its value for d between 0 and 4, and get your answer.

[–] 3 points4 points  (0 children)

The reason why no one has studied those problems in details is simply because no one has deemed them worthy of research.

Of course, whether a problem is interesting or not is subjective, but I can try to give my point of view. I would say that mathematicians usually tend to study problems that either:

1. Arise as sub-problems of larger problems that are already considered to be interesting, or;

2. Appear naturally from studying a structure or an object (e.g. when trying to generalise a result).

(I'm sure there are other reasons, but I think those two motivate most problems.)

The problems you are interested in, however, seem to be no more than brain-teasers. They do not appear as sub-problems of larger problems, and there does not seem to be any structure to their solutions. This could also simply mean that we have not yet found the underlying concept, but I doubt it.

Personally, I don't believe that there is a deeper structure here. For example, dealing with digits (which depend on the base ten representation) rather than integers indicates that a solution would not be algebraic. (For example, 13 = (2+63)/5 depends on the fact that the concatenation of the digits 6 and 3 represents the integer 63.) On the other hand, your solution also depends on the behaviour of algebraic operations which care not about your choice of base. This leads to a "contradiction" in the sense that you should expect a general solution to be both algebraic and not algebraic at the same time...

TL;DR: The problem is not useful and there's no nice structure to its solutions. This makes it uninteresting to study.

[–][S] 0 points1 point  (0 children)

Thank you so much! Every single concept from that book is completely new to me, so it's very helpful to have feedback to make sure I understand the material well.

[–] 0 points1 point  (0 children)

The set {z : |z| = 2} denotes the set of points with norm 2, which represents a circle centred at the origin of radius 2. The obvious parametrization is then simply 2e for 0 ≤ ϑ < 2π.

We can then use this parametrization to also find γ' = {f(2e) : 0 ≤ ϑ < 2π} = {(1+2e)2/4 : 0 ≤ ϑ < 2π} = {1/4+e+e2iϑ : 0 ≤ ϑ < 2π} which looks like this.

Note that f(γ) and γ' are used in part (c) of the question.

[–] 0 points1 point  (0 children)

As you said, this really depends on the definitions you are using. If this is in the context of a course, you should simply use the definition provided in class. The definition of integral domains I'm used to is the one provided on Wikipedia and doesn't mention anything about the existence of an identity. For this definition, the statement is certainly true.

[–] 1 point2 points  (0 children)

You have the right idea on how to do this, so here is a detailed solution. If at some point you feel like you can finish the problem by yourself, please try to do it.

You want to find for which x does

[; \frac{x^2+\frac{1}{x}}{x+\frac{1}{x}-1} = 1 ;]

so you can multiply the numerator and the denominator by [; x ;] to get

[; \frac{x^3+1}{x^2+1-x} = 1. ;]

Then, multiply both sides by the denominator

[; x^3+1 = x^2-x+1 ;]

and bring back everything on one side

[; x^3 - x^2 + x = 0. ;]

You can then factor the left-hand side by pulling out a factor of [; x ;] to get

[; x\cdot(x^2 - x + 1) = 0. ;]

The quadratic equation itself has no real roots, so you can see then that [; f(x) = 1 ;] only when [; x = 0 ;].

I want to note that, technically, [; f ;] is not defined at [; x = 0 ;] (because of the [; 1/x ;] terms) which means that [; f^{-1}(1) ;] is not defined either. The correct way to define [; f ;] for every real number is

[; f(x) = \frac{x^3+1}{x^2-x+1}. ;]

[–][S] 0 points1 point  (0 children)

This clears things up, especially with the "I look down upon you" example. If I understand correctly, one way to avoid ending the original sentence with a preposition would be with "This is the sort of English with which I will not put up," but then we lose the idiomatic meaning of "put up with" (as we did in your example with "look down upon").

Thank you for the clarification!

[–] 0 points1 point  (0 children)

There is a symmetry to these equations, so it's not too weird to try to find a solution while assuming that x, y and z are all equal. In this case, you get

x = y = z = (1+sqrt(1+4a))/2

or

x = y = z = (1-sqrt(1+4a))/2.

You can check that these work as solutions, though they might not be the only ones.

Finding sum of a series by in cheatatmathhomework

[–] 1 point2 points  (0 children)

Okay, first note that

Sum (5n-2)/((n-1)n(n+2)) = 5 * Sum 1/((n-1)(n+2)) - 2 * Sum 1/((n-1)n(n+2)),

where both series on the right are converging (from the comparison test with the p-series). So we'll evaluate the two series individually.

We start with the first series. By partial fraction decomposition, we find that

1/((n-1)(n+2)) = 1/(3(n-1)) - 1/(3(n+2)).

What you must see here is that this gives telescoping partial sums. To be more precise, the terms with denominator greater than or equal to 12 cancel out. You are left with

Sum 1/((n-1)(n+2)) = 1/3 + 1/6 + 1/9 = 11/18.

Now for the second one. We use the same technique and find that

1/((n-1)n(n+2)) = 2/(6(n-1)) - 3/(6n) + 1/(6(n+2)).

This again gives telescoping partial sums where all terms with denominator greater than or equal to 24 cancel out. You are left with

Sum 1/((n-1)n(n+2)) = (2/6 - 3/12) + (2/12 - 3/18) + (2/18) = 7/36.

Hence the series is equal to

Sum (5n-2)/((n-1)n(n+2)) = 5*(11/18) - 2*(7/36) = 8/3.

[–] 0 points1 point  (0 children)

So your average is the random variable X ~ N(21.1, 6.82/80). If we normalize this variable, we get Z = (X - 21.1)/(6.8/sqrt(80)) ~ N(0,1) and from a table for a normal variable, we find that

P(Z ∈ [-1.96, 1.96]) = 0.95.

Now if we simply rewrite the set inside the probability in terms of X we see that

Z ∈ [-1.96, 1.96]

iff (X - 21.1)/(6.8/sqrt(80)) ∈ [-1.96, 1.96]

iff X - 21.1 ∈ [-1.96*6.8/sqrt(80), 1.96*6.8/sqrt(80)]

iff X ∈ [21.1 - 1.96*6.8/sqrt(80), 21.1 + 1.96*6.8/sqrt(80)].

From this, you find that the constant you are looking for is

a = 1.96*6.8/sqrt(80) ≈ 1.49012.

In particular, 17 is outside of this interval since 17 < 21.1-1.5.

[–] 1 point2 points  (0 children)

Well the sample mean X_bar is normally distributed and both the mean and the standard deviation are known. This is true regardless of the sample size. Since you know the exact distribution, there is no reason to look for an approximation via the t-distribution. The t-distribution will be useful when you can only approximate your standard deviation with your sample.

You can then just calculate the probability of a normally distributed random variable with mean 98.2 and standard deviation 0.62/sqrt(19) to be less than 98.5. This gives you 0.982534.