That is the Edwards curve addition law, more info can be found here about the Edwards curve:
http://en.wikipedia.org/wiki/Edwards_curve
Everywhere I look it says the addition law takes 10M+1S+1D+7a, where M is field multiplication, S is field squaring, D is multiplying by a parameter d, and a...
Basically I'm writing my MSc dissertation right now, and I've been doing a lot on primes
I've written all my code in MATLAB, but my supervisor told me today that MATLAB is crap for higher order arithmetic and the primes only go up to something like 10 digits long.
SO I'm kinda screwed...
This is a question on a past exam I'm trying to asnwer.
So far I have, and I'm not sure if my understanding is correct yet:
Platonism:
Consider if man-kind became extinct, the statement 1+2=3 will always be able to be recovered.
Formalism:
3 is a consequence of the specific...
Basically I have a function file set up called BSform.m, I'm confident this has nothing to do with my problem
My problem is this driver file for the function
X=zeros(101,2);
for i = 0:0.01:1
[C1,P1]=BSform(200,200,2,0,0.02,i);
Y=[C1,P1]
X(100*i+1,1)=Y(1,1)
X(100*i+1,2)=Y(1,2)...
[Number Theory] Find all the ideals with the element 6 in them in Z[√-5]
Edited original question since I have now found the answer (I realize the title is inconsistent on the forum page), instead I am now trying to do part i) here
Is it possible to it this way:
Or is the structure of the...
I'm doing the last part of iii)
My lecturer has given a definition of the class number, abstract stuff and hasn't actually explained it in simple terms to get my head around.
In particular in any example when it comes to calculating the class number h(K) she just goes 'Hence h(K) = 2'...
For part i) I deduced via Dedekind's criterion that
<2> = <2,√6>2 & <3> = <3,√6>2
So ii) I am trying to do now, and my argument is thus:
Let a be an ideal in Z[√6]. Suppose that N(a) = 24.
By a proposition in my notes we have that
a|<24> = <2,√6>6<3,√6>2
so a = <2,√6>r<3,√6>s
for some r...
Ah ok, I'm not really sure how to work with the generators though.
I assume you mean something like
<2 , 1+ rt(-5)> = <4, 4(1+rt(-5)), 4> =<2>
Which I know its probably wrong but you get the idea. Can you give me a general rule on how to multiply together
<a, b + rt(c)><d, e +...
I'm doing part iii) here:
So far for p2 = <2> I have:
Show p2 is containd in <2>
Since [we have already shown]
p={r + s√-5 | r = s mod 2}
Then p2 elements are of the form (r+s√-5)2
= (r2 -5s2) + 2rs√-5
Since r&s have the same sign, then (r2 -5s2) is always even, & 2rs is always even too...
I need to calculate the norm of the ideal
p = (3, 1 - √-5)
All the information I have is that it's a prime ideal.
I managed to calculate the normal of the ideal q = (3, 1 + √-5) (which was 3) by finding a the determinant of a base change matrix by considering an integral basis
Here...
On the last step I need to be able to justify why we are removing the (-x) using the axioms only
If I had a lemma that said
y+z=x+z → y=x
then that way would be fine
But I don't, so essentially now I'm trying to prove this lemma
TA11 my lecturer mentioned
'Note that all function symbols have to be defined as total, and therefore 0−1 has to be assigned a meaning even though in reality the number 0 has no reciprocal; axioms TA8 and TA11 in effect define x−1 as the reciprocal of x when x ≠ 0, and 0 otherwise. This does...
y + (-x) = 0 → y = x
Need to do this in Tarski arithmetic, axioms below, as well as results below that which have been previously proved
I really have no idea how I can move this x over, can anyone help me out with a starting axiom?
Is it just TA4 that implies y=x? That simple? If it is...
Yeah we are advised to do this proof 'mathematically' rather than 'logically' (i.e not by natural deductions using rules of inference)
Here is an example proof, all the others have been done via the induction schema so I assumed this one was to be done that same way too
If not then I have no...
I have to do the following using these axioms PA1-7, the others below it are previously proved results I can use too.
[Sa] means the successor of a.
Base Case: y = S0
x.S0 = S0
→ x.0 + x = S0
→ 0 + x = S0
→ x = S0 & y=S0
Now the induction step is usually y=a to y=Sa, however this does...