## Problem 027

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Comments, questions and clarifications about PE problems.
jaap
Posts: 553
Joined: Tue Mar 25, 2008 3:57 pm
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### Re: Problem 027

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.
kenbrooker
Posts: 187
Joined: Mon Feb 19, 2018 3:05 am
Location: Northern California, USA

### Re: Problem 027

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
..."
jaap
Posts: 553
Joined: Tue Mar 25, 2008 3:57 pm
Contact:

### Re: Problem 027

kenbrooker wrote: Thu Dec 19, 2019 5:54 pmIf 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.
kenbrooker
Posts: 187
Joined: Mon Feb 19, 2018 3:05 am
Location: Northern California, USA

### Re: Problem 027

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
..."
kenbrooker
Posts: 187
Joined: Mon Feb 19, 2018 3:05 am
Location: Northern California, USA

### Re: Problem 027

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
..."
kenbrooker
Posts: 187
Joined: Mon Feb 19, 2018 3:05 am
Location: Northern California, USA

### Re: Problem 027

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
..."