Problem 027
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See also the topics:
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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.
_{Jaap's Puzzle Page}
 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
LucasC would too...
Happy Holidays
"Irregardless"
If you'd care to elaborate on such a simple relationship @ PE.net,
I would sure appreciate it, as I imagine
LucasC would too...
Happy Holidays
"Irregardless"
"Good Judgment comes from Experience;
Experience comes from Bad Judgment..."
Experience comes from Bad Judgment..."
Re: Problem 027
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 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
LucasC would too...
_{Jaap's Puzzle Page}
 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 preaddressing LucasC's question!!
Happy Polynomial Holidays,
glasshopper
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 preaddressing LucasC's question!!
Happy Polynomial Holidays,
glasshopper
"Good Judgment comes from Experience;
Experience comes from Bad Judgment..."
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 xaxis,
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...
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 xaxis,
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..."
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 LucasC'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!!
LucasC I sent you a PM...
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!!
LucasC I sent you a PM...
"Good Judgment comes from Experience;
Experience comes from Bad Judgment..."
Experience comes from Bad Judgment..."

 Posts: 1
 Joined: Tue Jun 21, 2011 5:23 pm
Re: Problem 027
Part of this problem states "The product of the coefficients, −79 and 1601, is −126479." Does this have any relevance whatsoever to the problem or the solution? It sure seems not.
ETA: Oh, never mind, I see at the end where one is to find the product of the coefficents of the appropriate polynomial to enter this product as the numerical answer. The statement above didn't make sense by itself.
ETA: Oh, never mind, I see at the end where one is to find the product of the coefficents of the appropriate polynomial to enter this product as the numerical answer. The statement above didn't make sense by itself.