The question concerns determining the speed, V, around the circular track of radius R. A simpler version of the problem is the situation when the container is accelerated in a straight line at acceleration a.
Excellent PbuK! I did miss that one - either I did not go far enough to find 4969, or did not use close enough tolerances. I did revise my algorithm to use loops based on only prime factors (although even there I did not go up to a factor as high as 4969).
I think this put the problem to rest.
MATLAB does have a nifty symbolic toolbox, so one can use it to add a series of fractions symbolically and
get the result as a ratio of integers. That's what I used to verify my first 8 "cousins", not "floating point arithmetic." I suppose one schooled in computer algebra (not my forte) could...
For n = 6, k has to be .GE. 1 and .LE. to 6. Hence, the moniker "6th cousin."
Let's look at "6th cousin" #1, namely 14, which yields a RHS of 4/7, so k = 4 which satisfies 1 =< k =< 6.
OK, so - at least as it was in the old days - 14 has the prime factors 2 and 7, so
1/2+1/7-1/14 = 7/14 +...
Thank you for your insightful comments. Sorry about the Excel spreadsheet. It turns out that 1722 is indeed one of the regular Giuga numbers, but not a 6th cousin. I actually let MATLAB cycle with a stride of 1 through and found the first eight 6th cousin values listed in my table. I have...
Sir: Hooray for the Kansas City Chiefs! Not the result I'd hoped for, but WOW, what a game!
Thanking you once more for your patience. I need to say that n = 6, representing the "6th Giuga cousin."
So, the RHS becomes k/(n+1) = k/(6+1) = k/7. This means that when we do find an X, who with its...
I wish it were so simple. OK, I took a look at your equations a few messages ago and need to point out that the "-1/n" term should be "-1/X," where big X is the product of the primes p1,p2,p3,...pm, and over to the left side where you have "n = p1p2p3...pm" it should read "X = the product of the...
We're rooting for the Bucs, given we are in Boston, MA now, and followed Tom Brady's successes here with the Pats. But those Rams look very healthy and good, so IDK how well Brady will be able to do with so many players knocked out. I hate post-season games where it's a blow-out.
Thanks very much. About to watch the Buc-Rams football game, and I have only skimmed your message, but FYI, I have attached an Excel spread sheet of the "cousins" I have found so far, using brute force searching with MATLAB. I will take a look at your suggestions to see what can be done.
Thank you. You are "preaching to the choir." In engineering we're often happy if we get within 10% of the answer, LOL. I remember when I reported to work at a Navy lab (which no longer exists) in 1976, one of my colleagues had just returned from a mathematics conference where they announced the...
Thank you for your response. From what I can tell - and G. H. Hardy's classic work "An Introduction to the Theory of Numbers" echoes your sentiment - there are many conjectures about the primes that are so easy to state, yet so hard to prove. I don't think my "Cousin" problem is any more than...
Why? I suppose the same question could be asked of any number theory inquiry, or for that matter,
and attempt to climb an unscaled mountain. "Because it's there!" It is interesting that nowhere in my
collection of number theory books are Giuga Numbers mentioned.
I should mention that in my...