Integer sum of powers Prove that for integers $a,b$ number : $(a+\frac{1}{2})^n+(b+\frac{1}{2})^n$ is integer for finitely many integers $n$.   I tried to use Binomial theorem but I didn't find any interesting.
How do I show that the sum $(a+\frac12)^n+(b+\frac12)^n$ is an integer for only finitely many $n$? Show that if $a$ and $b$ are positive integers, then $$\left(a +\frac12\right)^n + \left(b+\frac{1}{2}\right)^n$$is an integer for only finitely many positive integers $n$.   I tried hard but nothing seems to work. :(
Which is better: Employer-sponsored HSA or opening an HSA individually? So I know that:   Employer-sponsored HSA's allow you to deduct contributions directly from your paycheck, pre-tax With individual HSAs, you must contribute using post-tax dollars, then deduct those contributions from your taxable income when you do your taxes   Given those facts, I guess it's marginally better to use an employer-sponsored HSA because it means the government doesn't temporarily hold on to your taxed income, only to return it later after you do your taxes; you could theoretically put that money to work in an investment vehicle instead of having the government hold it hostage for a while.  But are there any other benefits (either way) that I'm not aware of? The  kind of threw me with the following statement:     The main advantage of making pre-tax contributions is the Federal Insurance Contributions Act tax (FICA) and Medicare Tax deduction, which amounts to a savings of 7.65% each to the employer and employee (subject to limits of the Social Security Wage Base).   But I'm not really sure what that's telling me.  My employer is offering an HSA, but is not offering to contribute to the HSA or match my contributions. So if it's all pretty much the same, I'd rather use the bank of my choosing and manage it privately.  Thanks. Android does not download any files from untrusted CA certificate site While browsing Stack Overflow, I noticed that I lost a lot of reputation:        Is this serial downvoting? Will it be corrected by a rep-recalculation? How often are those done?    Here's a second instance: Why is variance squared? I have found this figure of a mode of operation using a triple AES with m||IV+ctr input:   I suppose it's a custom mode of operation since I haven't seen it anywhere else. My question is if it is IND-CPA, and how to prove it in an adversarial model? I have a (symmetric) matrix M that represents the distance between each pair of nodes. For example,       A   B   C   D   E   F   G   H   I   J   K   L A   0  20  20  20  40  60  60  60 100 120 120 120 B  20   0  20  20  60  80  80  80 120 140 140 140 C  20  20   0  20  60  80  80  80 120 140 140 140 D  20  20  20   0  60  80  80  80 120 140 140 140 E  40  60  60  60   0  20  20  20  60  80  80  80 F  60  80  80  80  20   0  20  20  40  60  60  60 G  60  80  80  80  20  20   0  20  60  80  80  80 H  60  80  80  80  20  20  20   0  60  80  80  80 I 100 120 120 120  60  40  60  60   0  20  20  20 J 120 140 140 140  80  60  80  80  20   0  20  20 K 120 140 140 140  80  60  80  80  20  20   0  20 L 120 140 140 140  80  60  80  80  20  20  20   0   Is there any method to extract clusters from M (if needed, the number of clusters can be fixed), such that each cluster contains nodes with small distances between them. In the example, the clusters would be (A, B, C, D), (E, F, G, H) and (I, J, K, L).  I've already tried UPGMA and k-means but the resulting clusters are very bad.  The distances are the average steps a random walker would take to go from node A to node B (!= A) and go back to node A. It's guaranteed that M^1/2 is a metric. To run k-means, I don't use the centroid. I define the distance between node n cluster c as the average distance between n and all nodes in c.  Thanks a lot :) Can I travel within the US with an expired visa? I am an international student and authorized to stay in US for 2 years on my I-20 form. My visa expired on June this year and I need to fly to some states outside Georgia to visit colleges. Can I fly within the US?  If I can, what kind of document will the airport check when I depart? Will they notice my expired visa and turn me in to Homeland Security? In other words, will I get into trouble? 32 bit or 64 bit OS on a 512 mb VPS? When adding NOT NULL columns with a DEFAULT value - does PostgreSQL optimize this operation?  In case the table has n rows, a not-optimized alter-table-add-column would yield n writes of the default value - which could be very painful, obviously. With the optimization the DB would instantaneously create the new column, store just one copy of the default value which would be returned when no non-default value is found for that column in a suitable index data structure.  For example  such an . Which equals operator (== vs ===) should be used in JavaScript comparisons? I'm using  to go through JavaScript, and it's returning many suggestions to replace == (two equals signs) with === (three equals signs) when doing things like comparing idSele_UNVEHtype.value.length == 0 inside of an if statement.  Is there a performance benefit to replacing == with ===?   Any performance improvement would be welcomed as many comparison operators exist.  If no type conversion takes place, would there be a performance gain over ==? Light emission from the vicinity of a black hole The photograph of the black hole, at the centre of galaxy M87, is now a part of human history. We are told that the red-colour is caused by hot gas, matter &amp; radiation around the "edge", the Event Horizon. Given the power of the gravitational forces emanating from black hole, why aren't these three things drawn into the abyss? If light is bent/ drawn into the BH, how does this red image arise? The logic, here, would be that a body, on the Event Horizon, is immune from the gravity--which sounds pretty fatuous? In what order are the dated ordered by? Certainly not alphanumeric order.  ls -lt sorts by modification time. But I need creation time. Does mass distribution/center of mass affect free fall? An everyday is example would be cars. If you were to drop a car, not taking air resistance into account, would a front-heavy car tumble forward as it fell? And a car with a perfect 50/50 front/rear weight distribution fall straight? Does mass distribution/center of mass affect free fall? Or would they both fall straight down without tumbling? Can you explain why please? I am trying to testfor a player holding a diamond sword named Notch's Legendary Sword. How can I do that? Who ever gets the best answer will be able to get the first look at my NEW adventure map! Seeking SF Short Story from late 1970s/early 1980s about dystopian future with an old man playing a piano I read this in the late 1970s or early 1980s, possibly as part of a juvenile fiction anthology/collection. From what I remember in this story a post-apocalyptic society requires each member to go out before their 18th birthday and kill an "oldster" adult (i.e. I think over 30 similar to the time in Logan's Run) to make room for themselves.  Otherwise they will be killed.   The kids actually cut off the hand of the victim which has a tattoo on it with the birthdate/age. In this case, the world has gotten to the point the person who kills the oldest person gets a "prize" of one more year of life beyond 30. Note, this is NOT Logan's run nor the novel upon which it is based.   The hero in this case finds an old man, the oldest he had ever seen (something like 45-50 I think) making wonderful sounds with a strange device. It was a piano.   He then confronts the old man  SPOILER ALERT     ... but cannot bear to kill him. The old man takes pity on the boy and offers him his hand to take back so he can live. The boy asks if the old man will still be able to play the piano with just one hand. The old man smiles and says "Yes, but not as well. " Then, the boy decides to stay with the old man and risk becoming an outlaw so he can learn to play the piano. How to exclude files from Visual Studio compile? I'm in the process of refactoring a project.  I've got an entire subfolder which is known to be broken.  Is there any declarative way to exclude that folder from the compile temporarily while I test the refactoring thus far?  I realize I could delete the folder, but I'd like to do this through configuration if possible. Evaluate  $$\lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$$ Prove $\binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a}$ I want to prove this equation, $$ \binom{n}{a}\binom{n-a}{b-a} = \binom{n}{b}\binom{b}{a} $$ I thought of proving this equation by prove that you are using different ways to count the same set of balls and get the same result. But I'm stuck. Help me please.   (Presumptive) Source: Theoretical Exercise 1.14(a), P19, A First Course in Pr, 8th Ed, by S Ross. Is it possible to add a list of hosts that are only specific to a certain user? Perhaps a user-specific hosts file?  This mechanism should also complement the entries in the /etc/hosts file. Posts flagged for attention but there are none The comment with the Flagged Posts that seems to be new as of today states there are posts with flags awaiting my attention but the counter shows zero and when I go into flagged posts there is nothing waiting for me.