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Undergrad Question help

Could someone post a solution to this please?
I got as far as showing that the floor function (2+root5)^p - 2^(p+1) is always even.
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(edited 8 years ago)
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Reply 1
Avatar for TeeEm
TeeEm
19
Original post by kprime2
Could someone post a solution to this please?

Attachment not found


would love to help but I do not do any pure
Most of the purists have left TSR !!
Reply 2
Avatar for Zacken
Zacken
22
Original post by kprime2
Could someone post a solution to this please?
I got as far as showing that the floor function (2+root5)^p - 2^(p+1) is always even.
Attachment not found


Write N=(2+5)p+(2−5)pN = (2 + \sqrt{5})^p + (2 - \sqrt{5})^p, can you see that this is an integer?

Can you use this combined with the size of (2−5)p(2-\sqrt{5})^p (hint:

Spoiler

to show that N=[(2+5)p]N = [(2 + \sqrt{5})^p] ?

Now in the two binomial expansions, look at the binomial coefficients, (pk)\binom{p}{k} What is this divisible by? So all the terms in the binomial expansions except the first and last term are divisible by that thing. What can you say about the first term, though?

Conclude that N≡2⋅2p(modp)N \equiv 2 \cdot 2^p \pmod{p}.
(edited 8 years ago)
Reply 3
Avatar for kprime2
kprime2
OP
20
Original post by Zacken
Write N=(2+5)p+(2−5)pN = (2 + \sqrt{5})^p + (2 - \sqrt{5})^p, can you see that this is an integer?

Can you use this combined with the size of (2−5)p(2-\sqrt{5})^p (hint:

Spoiler

to show that
Unparseable latex formula:

N = \floor{2 + \sqrt{5}} = 0

?

Now in the two binomial expansions, look at the binomial coefficients, (pk)\binom{p}{k} What is this divisible by? So all the terms in the binomial expansions except the first and last term are divisible by that thing. What can you say about the first term, though?

Conclude that N≡2⋅2p(modp)N \equiv 2 \cdot 2^p \pmod{p}.


Woah wtf haha, I swear you're like an A Level student. Impressive though, shall have another try later.
Reply 4
Avatar for Zacken
Zacken
22
Original post by kprime2
Woah wtf haha, I swear you're like an A Level student. Impressive though, shall have another try later.


I made some mistakes that I've edited now. :colondollar:

Edit: tbf so is physicsmaths and you tagged him :tongue:
Reply 5
Avatar for kprime2
kprime2
OP
20
Original post by Zacken
I made some mistakes that I've edited now. :colondollar:

Edit: tbf so is physicsmaths and you tagged him :tongue:


Haha I meant it as a compliment. What I meant was I'm assuming you're doing STEP or something, as I wouldn't expect an A-level student to tackle a question of that magnitude. But cheers for the help, sounds right.

I swear physmath did like STEP I-III or some **** so I thought he'd knew. :P
Reply 6
Avatar for Zacken
Zacken
22
Original post by kprime2
Haha I meant it as a compliment. What I meant was I'm assuming you're doing STEP or something, as I wouldn't expect an A-level student to tackle a question of that magnitude. But cheers for the help, sounds right.

I swear physmath did like STEP I-III or some **** so I thought he'd knew. :P


I did take it as a compliment! :smile: Yeah, I've got a Cambridge offer, so I'll be doing STEP I-III as well. :yep:
Reply 7
Avatar for kprime2
kprime2
OP
20
Original post by Zacken
I did take it as a compliment! :smile: Yeah, I've got a Cambridge offer, so I'll be doing STEP I-III as well. :yep:


Amazing. Well done!
Reply 8
Avatar for Zacken
Zacken
22
Original post by kprime2
Amazing. Well done!


Thanks. :colondollar:

Let me know how you get on with the question and whether my approach works or not. :biggrin:

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