steiner1745

Hi, all!

Here's a slightly altered question I found in another group.

I've mulled it over for a while, but I'm stuck. Help!

The question is

Do there exist a Pythagorean triple (a,b,c)

and a positive integer D (not a square)

such that

a^2 - Db^2 = 1? (1)

So far I have that

c^2 = a^2 + b^2

so

c^2 - (D+1)b^2 = 1 (2)

How can I solve the simultaneous Pell

equations (1) and (2)?

Any ideas?

I was trying to decipher

http://answers.yahoo.com/question/index?qid=201305...

when I ran across the pretty song

yannna oba yanna ath wela.

What is the translation of the title?

The language here is Sinhala,

which is spoken in Sri Lanka.

(Adapted from the latest issue of THE BENT)

You are floating in a sea of 8's on a raft

with number 101. You discover that you can take

an 8 and insert it into your raft to enlarge it.

Thus your starting move can be 8101, 1801, 1081 or 1018.

Unfortunately, every time you do this, the raft divides itself

by its smallest prime factor. If your raft goes below 100

it sinks. What is the maximum of insertions you can

make before you sink?

Note: 1 is not a prime.

b). Work the same problem with 7 instead of 8

(original problem).

1. How do you pronounce Hungarian é?

Books and pronunciation guides say to pronounce

it like the ay in day.

But on Livemocha they were pronouncing it like long e.

The audio on Google Translate also pronounces it like long e.

So, for example, is the word hét(seven) pronounced more like

English hate or heat?

2. I was trying to translate There are rats in your mother's house

to Hungarian.

After looking at many sources, I came up with

Az anyad házában patkányok vannak.

Is this a correct translation?

Thanks in advance!

I'm trying to draw a circuit diagram with

a battery and some resistors so I can ask

a couple of questions in this forum.

I was playing with a previous question involving finding

the arclength of the secant curve from 0 to π/4

(http://answers.yahoo.com/question/index?qid=201203...

and ran into the following integral:

∫ (2cos^2 x -1) dx /√(cos^4 x - cos^2 x + 2).

The Wolfram integrator gives the answer as

arctan(√2 sin 2x/√(cos 4x + 15).

Verify this by integration.

Of course, one could differentiate the answer,

but the question is: How does one arrive at it?

Another way of writing the answer is

arctan(sin x cos x/√(cos^4 x -cos^2 x + 2)

Any ideas?

We've had some really excellent questions concerning elliptic curves

posed by Rita the Dog, Dragan and others. Here's one more:

Which Fibonacci numbers are magic square numbers, i.e.

of the form 1/2(n^3+n)?

To reduce this to elliptic curves let's use the identity

L_k^2 -5F_k^2 = 4(-1)^k.

where L_k is the kth Lucas number.

Plugging (n^3-n)/2 into this identity, we get

(2L_k)^2 = 5(n^6+2n^4+n^2) + 16 if k is even

(2L_k)^2 = 5(n^6+2n^4+n^2) - 16 if k is odd. (*)

Right now, I'd like to assume k is odd and solve

that case first. If k is odd, there are 3 known integer solutions to (*)

namely n = 1,2,4. My question is: Are there any others?

To reduce this to elliptic curves, let w = 2L_k, m = n^2. We get

w^2 = 5m^3 + 10m^2 + 5m -16.

Finally, let m = x/5 and simplify: We get

y^2 = x^3 + 10x^2 + 25x -400 (**)

on setting y = 5w.

Question 1.There are 3 positive integer solutions for (**)

(5, 10), (20, 110) and (80,760)

are there any others?

I ran (**) through PARI and found

that it has trivial torsion, but I couldn't

find a system of generators for the Mordell-Weil

group because the conductor is 45200

and PARI didn't have a data file for this curve.

Question no. 2. What is the rank and a system

of generators for (**)?

Any help would be much appreciated!

I got an e-mail from Yahoo Award notifications,

UK-Ireland that said I won 720 English pounds.

Is this real or is it just another scam?

What is the meaning of the last word of the phrase

В глибокі хущі?

It's from another question and the language

is Ukranian.

I couldn't find the word in any of the online

Ukranian dictionaries.

Thanks.

This is a follow-up to John D's question about a number that can

be doubled and switched around.

Show that for any integer m >1 there are positive integers n, m such

that if we multiply n by m we get n with the last digit

brought to the front. You may suppose that (n,10) = 1.

Hint: Look at the repeating part of the decimal expansion of 1/n.

Finally, prove that your final conjecture is true.

Examples. 1/7 = 0.142857...

142857*5 = 714285.

1/13 = 0.076923...

076923*4 = 307692.

1/19 = 0.052631578947368421...

0526315789473684421*2 = 105263157894736842.

(Note: You must retain the leading zeros of 1/n for this

pattern to work.)

Finally,

1/29 = 0.0344827586206896551724137931...

and 0344827586206896557724137931*3 =

1034482758620689655172413793.

There you have examples for m = 2,3,4 and 5.

Show how you can find such an example for any m

and prove your result is correct.

Rita's last problem was excellent.

Here are 2 more in the same vein.

Let S_k(n) be the sum of the kth

powers from 1 to n.

The following are known

S_1(n) is a square for infinitely many n.

S_2(n) is a square only for n = 1 and 24.

(This is Lucas' problem of the square pyramid.

See www.math.ubc.ca/~bennett/paper21.pdf

for a nice discussion of this and further problems.)

S_3(n) is always a square, since it is S_1(n)^2

My questions are as follows:

1). Show that S_5(n) = n^2*(n+1)^2*(2n^2+2n-1)/12

is a square for infinitely many n.

(Hint: Reduce the problem to solving a Pell equation.)

2). When can S_4(n) = n*(n+1)*(2n+1)*(3n^2+3n-1)/30

be a square?

Consult

http://www.math.rutgers.edu/~erowland/sumsofpowers...

for more information on sums of consecutive powers.

Here's a bit of text:

Cai rer got mitzon tesfe, larour twoch nas sultetc, wirti ladfuen freinu nadiar tainkeep raer planag.

A headline was

Halbas cadmil juilfut.

Any ideas?

Every day I get some news in Spanish from

Bogota and they always have the phrase

Pico y placas hoy...

What would be a good translation of this idiom?

I know the individual words, but what does this

phrase mean idiomatically?

¡Muchísimas gracias!

Hi, all!

I recently read that Colombia was building a network of highways

called "Transversal de las Américas". Is this project still on track

or has it been cancelled?

The following question has been posted many times on YA:

A school has n lockers in one hallway and n students are assigned one each.

All lockers are initially closed.

The first student is instructed to walk down the hall and open every locker. The second student is instructed to walk down the hall and close every other locker. The third student is instructed to walk down the hall and change the state (if open, close it or if closed, open it) of every third locker. The fourth student would then change the state of every fourth locker, etc. What lockers would be left open after all n students are done?

We know that only the squares <= n are open after all the operations are done.

My question is: Find a sequence of operations which leave only

the cubes after all n operations are done.

BTW I first saw the locker problem on the 1967 Putnam

exam. They also asked the question about the cubes,

but never published an answer for it.

Find the exact value of the following integral.

It appeared as a problem on the qualifying

exam for this year's MIT integration bee.

∫[0..π/2] sin(8x^3)*cos(x) dx.

Note: The indefinite integral is probably

impossible to calculate. The Wolfram

integrator could not find a formula for it.

Here are two questions related to a seemingly

intractable problem I posed a while ago.

The intractable problem is:

Does there exist an odd whole number n

in base 10 such that

a). n is a square

b). The only digits of n are 0 and 1?

We now know there is no such number less than 10^48.

Here are the related problems.

One is easy, the other as hard as the first one.

1). Does there exist a whole number n in base

10 such that

a). n is a square

b). The only digits of n are 1 and 3?

2). Here are 2 near misses to the

intractable question.

1049^2 = 1100401

375501^2 = 141001001001.

Are there any other examples of this sort?

That is, is there another such square

with all 0's and 1's with the exception

of 1 rogue digit?

Let p(x) = x^(2n) + x^n + 1.

Prove or disprove the following conjecture:

p(x) is irreducible if and only if n = 3^k,

where k is a nonnegative integer.

The following question came up in a

problem someone asked me.

Evaluate the infinite product

Π(n=2..∞)n^2/(n^2-1).

Using PARI and computing 100000 terms makes it almost certain that

the answer is 2, but how does one prove it?

The following 4 problems were proposed by "Buddhist" of the Google

group sci.math about 8 years ago, but were never solved satisfactorily.

Let S_k(n) = 1^k + 2^k + ... + n^k.

The question is: When is S_k(n) a square?

If k = 1, we know that S_1(n) = n(n+1)/2 and this can be

converted to a Pell equation, which has infinitely many solutions.

Thus there are infinitely many triangular numbers which are squares.

If k =2, S_2(n) = n(n+1)(2n+1)/6.

This is Lucas's problem of the square pyramid and it is known

that S_2(n) is a square only for n = 1 and n = 24.

If k = 3, S_3(n) = S_1(n) ^2, so S_3(n) is always a square.

The 4 problems are: Find all n for which

a. S_4(n) = n(n+1)(2n+1)(3n^2+3n-1)/30 = m^2

b. S_5(n) = n^2*(n+1)^2*(3n^2 + 2n-1)/12 = m^2

c. S_6(n) = n(n+1)(2n+1)(3n^4 + 6n^3 -3n + 1)/42* = m^2

d. S_7(n) = n^2*(n+1)^2*(3n^4 + 6n^3 -n^2 -4n+2)/24 = m^2.

The solution of any of these 4 questions will certainly

earn best answer and would be most welcomed.

Note: These are 4 separate problems.