1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
igor_vitrenko [27]
3 years ago
8

Find two positive nontrivial solutions of x^2-Ny^2=1 when N is 15, 7, or 99

Mathematics
2 answers:
Pavel [41]3 years ago
8 0

Hello,


if n=15: (4,1),(31,8),(244,63)


if n=7: (8,3),(127,48),(2024,765)


if n=99: (10,1),(199,20),(3970,399)


(Pell 's equation)


Download xls
KengaRu [80]3 years ago
3 0

The solutions to the Pell equation

... x^2 -N·y^2 = ±1

are (x, y) = (p, q) where p/q are alternate convergents‡ of the continued fraction of √N.


The continued fractions for √15 and √99 can be written as

... √15 = 3 + 1/(1 + 1/(6 + 1/(1 + 1/(6 + 1/...))))

or, more compactly, as {3, {1, 6}};

and

... √99 = 9 + 1/(1 + 1/(18 + 1/(1 + 1/(18 + 1/...))))

or, more compactly, as {9, {1, 18}}.


The compact form of the continued fraction for √7 is {2, {1, 1, 1, 4}}.


In all cases, the continued fraction terms of a square root repeat after some finite-length sequence. In the compact form, the repeating sequence is shown in curly brackets. This permits the convergents to be defined recursively.


Successive convergents of the square root are p[k]/q[k], where

... p[k] = a[k]·p[k-1] +p[k-2] . . . . where p[-1]=0, p[0]=1

... q[k] = a[k]·q[k-1] +q[k-2] . . . . where q[-1]=1, q[0]=0

and the a[k] are written in the compact form as {a[1], {a[2], a[3], ...}}


For the given numbers N, the (x, y) values that are solutions to the given equation can be defined recursively quite nicely. The same recursive equation (written here as pq[ ]) is used for both p and q (as above), but with different initial conditions for p than for q.


For N=15, we can write

... pq[k] = 8·pq[k-1] - pq[k-2] . . . . . {p[-1], p[0]} = {4, 1} . . . {q[-1], q[0]} = {-1, 0}

so {x, y} = {4, 1}, {31, 8}, {244, 63}, {1921, 496}, ...


For N=99, we can write

... pq[k] = 20·pq[k-1] - pq[k-2] . . . . . {p[-1], p[0]} = {10, 1} . . . {q[-1], q[0]} = {-1, 0}

so {x, y} = {10, 1}, {199, 20}, {3970, 399}, {79201, 7960}, ...


For N=7, we can write

... pq[k] = 16·pq[k-1] - pq[k-2] . . . . . {p[-1], p[0]} = {8, 1} . . . {q[-1], q[0]} = {-3, 0}

so {x, y} = {8, 3}, {127, 48}, {2024, 765}, {32257, 12192}, ...


_____

‡ Convergents alternate between being larger than the square root and smaller than the square root. However, when there are more than 2 values in the repeating sequence of the continued fraction, the convergents that result in a solution to the Pell equation may be farther apart than every other convergent. For N=7, every 4th convergent gives a solution.

You might be interested in
At Dorthys hair salon, 4 gallons of shampoo will last 3 weeks. She has 7 gallons of shampoo. How much more must she order if she
lara31 [8.8K]
X=number of gallons that she must order.

4 gallons-----------------3 weeks
(7 gallons + x)--------12 weeks

We solve this rule of 3.
(7 gallons +x)3 weeks=4 gallons*12 weeks
21 gallons weeks+3 x weeks=48 gallons weeks
3x weeks=48 gallons weeks - 21 gallons weeks
3x weeks=27 gallons weeks
x=(27 gallons weeks) / 3 weeks=9 gallons.

Solution: she must order 9 gallons more.
5 0
3 years ago
A store charges a restocking fee for any returned item based upon the item price. An item priced at $200 has a fee of $12. An it
Vesna [10]

Divide the restocking fee by the price of the item:


12/200 = 0.06

9/150 = 0.06

Multiply by 100 to get the percent:

0.06 x 100 = 6%

Answer: 6%

3 0
2 years ago
ANGLE KLM and ANGLE RST are supplementary angles. If ANGLE KLM is (7x) and ANGLE RST is (36-x) what is the value of x ? Please h
liberstina [14]

Answer:

30

Step-by-step explanation:

Angle B measures 60°. What is the measure of the angle that is complementary to angle B?

30°

60°

120180°

4 0
2 years ago
24 dived by 7=what is the remainder
motikmotik
3 is the remainder because the closest is 7 times 21 so it’s remainder 3
7 0
3 years ago
Read 2 more answers
Which is the most effective brand name shoe. is it Experimental vs. Observational
balandron [24]

Answer:

nike its observational

Step-by-step explanation:

6 0
2 years ago
Other questions:
  • Taylor graphs the system below on her graphing calculator and decides that f(x)=g(x) at x=0, x=1, and x=3. Provide Taylor some f
    15·1 answer
  • Kevin's collection has 2 times as many dolls in it as Mark's collection. Together, they have 93 dolls. How many dolls does Kevin
    8·1 answer
  • Is the sequence geometric ? -48,96,-192,384...
    14·1 answer
  • A
    5·1 answer
  • Which side of the polygon is exactly 6 units long?
    14·1 answer
  • A certain type of golfball is tested by a robot that hits the ball with a consistent impact force. The distances this type of ba
    12·1 answer
  • 28)Hace un año compré una finca que se ha revalorizado el 20% (es decir que el precio de la finca hoy es el 20% más caro que su
    14·1 answer
  • 2x + 3x = _____??<br><br> help me pls
    6·1 answer
  • Im in 7th grade and Im failing integers
    5·1 answer
  • You have a standard deck of 52 playing cards. you draw 2 cards without replacement. which action, performed before the draws, in
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!