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• I was looking at http://www.primepuzzles.net/problems/prob_037.htm and in the proof of the formula mr Fengsui claims that ...the class of residues Tn mod mn
Message 1 of 5 , Feb 28, 2003
I was looking at

http://www.primepuzzles.net/problems/prob_037.htm

and in the proof of the formula mr Fengsui claims that
"...the class of residues Tn mod mn is equivalent to the class of
residues Tn+<0,1,2,...,pn=1>*<mn> mod m[n+1],"

could anyone explain to me in simple terms why this is?

any help will be greatly appreciated. have a good day.
• Suppose that: the Tn is a complete set of residues prime to mn, the least number more than 1 in this set U(Tn) is the n-th prime pn. The number of elements of
Message 2 of 5 , Feb 28, 2003
'Suppose that: the Tn is a complete set of residues prime to mn, the least
number more than 1 in this set U(Tn) is the n-th prime pn. The number of
elements of the set Tn is | Tn |=(p1-1)*(P2-1)*...*(p[n-1]-1). If p>p[n-1]
is a prime, then p belongs the class of residues Tn mod mn.'

I don't get this. mn is defined as "Let mn=p0*p1*...p[n-1]", therefore
m2=2.3=6 and m3=2.3.5=30

However, the number of residues prime to 30 is
|{2,3,5,7,11,13,17,19,23,29}|=10, and not the 8 predicted by Liu. I suppose
the p>3 property comes into play...

'If p>p[n-1] is a prime, then p belongs the class of residues Tn mod mn.'

Suppose p is a prime and p doesn't belong to tmod30, this is impossible.
Induction carries on from here.

I think what '"...the class of residues Tn mod mn is equivalent to the class
of
residues Tn+<0,1,2,...,pn=1>*<mn> mod m[n+1],"'

means is that we are looking at either (pn,mn)=1 OR (tn+pn,mn)=1.

HTH

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry/maths/
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-----Original Message-----
From: velozant <velozant@...> [mailto:velozant@...]
Sent: 28 February 2003 22:15

I was looking at

http://www.primepuzzles.net/problems/prob_037.htm

and in the proof of the formula mr Fengsui claims that
"...the class of residues Tn mod mn is equivalent to the class of
residues Tn+<0,1,2,...,pn=1>*<mn> mod m[n+1],"

could anyone explain to me in simple terms why this is?

any help will be greatly appreciated. have a good day.

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• ... Woh! Steady on, Jon. You re _way_ off base here. 2 is not coprime to 30. gcd(2,30)=2 3 is not coprime to 30. gcd(3,30)=3 5 is not coprime to 30.
Message 3 of 5 , Mar 1, 2003
--- Jon Perry <perry@...> wrote:
> 'Suppose that: the Tn is a complete set of residues prime to mn, the least
> number more than 1 in this set U(Tn) is the n-th prime pn. The number of
> elements of the set Tn is | Tn |=(p1-1)*(P2-1)*...*(p[n-1]-1). If p>p[n-1]
> is a prime, then p belongs the class of residues Tn mod mn.'
>
> I don't get this. mn is defined as "Let mn=p0*p1*...p[n-1]", therefore
> m2=2.3=6 and m3=2.3.5=30
>
> However, the number of residues prime to 30 is
> |{2,3,5,7,11,13,17,19,23,29}|=10, and not the 8 predicted by Liu. I suppose
> the p>3 property comes into play...

Woh! Steady on, Jon. You're _way_ off base here.

2 is not coprime to 30. gcd(2,30)=2
3 is not coprime to 30. gcd(3,30)=3
5 is not coprime to 30. gcd(5,30)=5

1 is coprime to 30. gcd(1,30)=1

|{1,7,11,13,17,19,23,29}| = 8 as correctly stated by Liu.

And Liu was not "predicting", this isn't reading chicken entrails, it's a
simple mathematical fact.

Phil

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• Woh! Steady on, Jon. You re _way_ off base here. 2 is not coprime to 30. gcd(2,30)=2 3 is not coprime to 30. gcd(3,30)=3 5 is not coprime to 30. gcd(5,30)=5 1
Message 4 of 5 , Mar 1, 2003
'Woh! Steady on, Jon. You're _way_ off base here.

2 is not coprime to 30. gcd(2,30)=2
3 is not coprime to 30. gcd(3,30)=3
5 is not coprime to 30. gcd(5,30)=5

1 is coprime to 30. gcd(1,30)=1

|{1,7,11,13,17,19,23,29}| = 8 as correctly stated by Liu.'

Correct. As the whole page in question
(http://www.primepuzzles.net/problems/prob_037.htm) is completely littered
with typos and misleading nomenclature, I don't feel completely aggrieved at
having made such a simple error.

As to what 'Tn mod mn is equivalent to the class of residues
Tn+<0,1,2,...,pn=1>*<mn> mod m[n+1]'

T_n mod m_n is equivalent to the class of residues
Tn+(<0,1,2,...,pn-1>*<mn>) mod m_(n+1)

which comes clear if you look at the examples, except for m_n is incorrectly
defined.

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry/maths/
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com
• Thanks a lot for your responses. What I am confused about is that I don t understand how adding * to Tn makes it equivalent mod m[n+1].
Message 5 of 5 , Mar 1, 2003
Thanks a lot for your responses. What I am confused about is that I
don't understand how adding <0,1,2,...,pn-1>*<mn> to Tn makes it
equivalent mod m[n+1]. What I would like is a theorem like the one
that one can uset to prove that if (k,S)=1 and S is a reduced
residue system then so is k*S. I think it should be obvious since
most people accept it and I see from examples that it is true, but I
just want to know what theorem he is using to imply this equivalence
or if it follows directly from the definition of Tn. Any help will
be greatly appreciated.
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