Problem 027

A place to air possible concerns or difficulties in understanding ProjectEuler problems. This forum is not meant to publish solutions. This forum is NOT meant to discuss solution methods or giving hints how a problem can be solved.
Forum rules
As your posts will be visible to the general public you
are requested to be thoughtful in not posting anything
that might explicitly give away how to solve a particular problem.

This forum is NOT meant to discuss solution methods for a problem.

In particular don't post any code fragments or results.

Don't start begging others to give partial answers to problems

Don't ask for hints how to solve a problem

Don't start a new topic for a problem if there already exists one


See also the topics:
Don't post any spoilers
Comments, questions and clarifications about PE problems.
User avatar
jaap
Posts: 549
Joined: Tue Mar 25, 2008 3:57 pm
Contact:

Re: Problem 027

Post by jaap »

All the solutions on a parabola are related to each other in a very simple way. Given that the solution to the problem lies on such a parabola, it would be a spoiler to discuss it too much.

User avatar
kenbrooker
Posts: 137
Joined: Mon Feb 19, 2018 3:05 am
Location: Oregon, USA

Re: Problem 027

Post by kenbrooker »

Thanks jaap...

If you'd care to elaborate on such a simple relationship @ PE.net,
I would sure appreciate it, as I imagine
Lucas-C would too...

Happy Holidays
"Irregardless"
"Good Judgment comes from Experience;
Experience comes from Bad Judgment
..."
Image

User avatar
jaap
Posts: 549
Joined: Tue Mar 25, 2008 3:57 pm
Contact:

Re: Problem 027

Post by jaap »

kenbrooker wrote:
Thu Dec 19, 2019 5:54 pm
If you'd care to elaborate on such a simple relationship @ PE.net,
I would sure appreciate it, as I imagine
Lucas-C would too...
Take a look at hk's post on the first page of the problem's discussion forum. It shows the relationship between two solutions. If you plot all the related solutions (i.e. for all p as used in his post) you get the parabola.

User avatar
kenbrooker
Posts: 137
Joined: Mon Feb 19, 2018 3:05 am
Location: Oregon, USA

Re: Problem 027

Post by kenbrooker »

Thanks Much jaap...

I will do that next, even at 1am here on the USA's West Coast; and,
my compliments to you for finding that and to
hk for pre-addressing Lucas-C's question!!

Happy Polynomial Holidays,
glasshopper
"Good Judgment comes from Experience;
Experience comes from Bad Judgment
..."
Image

User avatar
kenbrooker
Posts: 137
Joined: Mon Feb 19, 2018 3:05 am
Location: Oregon, USA

Re: Problem 027

Post by kenbrooker »

Speaking of PE27 and parabolas, came across this definition of a parabola's equation --

The standard form is (x - h)^2 = 4p (y - k), where the focus is (h, k + p) and the directrix is y = k - p.
If the parabola is rotated so that its vertex is (h,k) and its axis of symmetry is parallel to the x-axis,
it has an equation of (y - k)^2 = 4p (x - h), where the focus is (h + p, k) and the directrix is x = h - p.

Sure enough, there's hk right in the thick of it!
Maybe "Royalties" are in order?

Happy Parabolic Holidays to
hk and All other vertices by
first and last initials...
"Good Judgment comes from Experience;
Experience comes from Bad Judgment
..."
Image

User avatar
kenbrooker
Posts: 137
Joined: Mon Feb 19, 2018 3:05 am
Location: Oregon, USA

Re: Problem 027

Post by kenbrooker »

In summary of/to Lucas-C's challenge --

In a graph of a,b pairs such that prime P = N^2 + aN + b for -100 <= a <= 100 and 0 <= b <= 2000 and
such that the number of Ps for consecutive values of N is greater than 15, for example...
Why is an apparent parabola displayed?

I used the same graphical approach and found b = f(a), indeed a parabola...

Thanks to jaap and hk, I derived the same b = f(a) algebraically...

However, I don't see that finding b = f(a) -- a parabola -- explains
Why is b = f(a) a parabola?
Is that not at all...
Surprising???

Or, maybe I get it -- WHO CARES?! Maybe as moot as --
If you want a headlight with
parallel rays, use a
reflector that's...
Parabolic!!

Welcome 2020 --
THE Year of...
VISION!!


Lucas-C I sent you a PM...
"Good Judgment comes from Experience;
Experience comes from Bad Judgment
..."
Image

Post Reply