Adam Ledger

Mr. Adam Ledger

335 Reputation

11 Badges

4 years, 246 days
unemployed
hobo
Perth, Australia

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These are questions asked by Adam Ledger

As an example question, I am trying to learn a method for assertaining whether a Set with a given predicate is finite or infinite.

 

For example, is $S$ infinite or finite, if it is defined as:

 

$$S={\biggl\{k \in \mathbb N:\biggl\lfloor {\frac {{k-1}}{{2}
}} \biggr\rfloor-\biggl\lfloor {\frac {{\ln(\pi^{k})}}{{\ln(10)}
}} \biggr\rfloor=0}\biggr\}$$
 

Hi I was hoping if someone could mark this proof for a lemma regarding the Euler totient functin for me.

 

Thanks in advance.

 

totient_lemma_proof.mw

What is the difference between a divisor and a factor? Like:

with(numtheory):

X := 89733992396903316277681863138688595394562888838833;

S := map(expand, [op(ifactor(X))]):

And we have the singleton of unity for the output of:

divisors(X/(product(S[j], j = 1 .. nops(S))));

and so on:

seq(divisors(X/(product(S[j], j = 1 .. nops(S)-k))), k = 0 .. 3);

but yet if i try

divisors(X);

my computer freezes.

So yes although my friend is somewhat embarassed to not know the difference and needed me to ask for him what the difference between a divisor and a factor is. 

 

Like is it just the uniqueness? ie a factor is one of the terms in the product of a number's prime factorization, where as any number that divides the number is considered to be a divisor?

like for example i can then go:

seq(min(`minus`(divisors(X/(product(S[j], j = 1 .. nops(S)-k))), {1})), k = 1 .. 10);

X:=X/(131*(139*(151*(163*(173*167)*157)*149)*137)*127)

from the first 10 terms reducing it original number substantially to:

 17026820583257598495326242577

and carry this on i dont see why i cant end up with the divisor set that caused my computer to freeze if i just called divisors(X) as I originally did

 


 

https://www.youtube.com/watch?v=qSO2lLiDybg

with(numtheory):

X := 89733992396903316277681863138688595394562888838833;

89733992396903316277681863138688595394562888838833

(1)

S0 := map(expand, [op(ifactor(X))]);

[31, 47, 59, 61, 67, 71, 73, 79, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173]

(2)

P := convert([seq(min(`minus`(divisors(X/(product(S0[j], j = 1 .. nops(S0)-k))), {1})), k = 1 .. 15)], '`*`');

72255473008151934648648475807453

(3)

S1 := divisors(X/P):

seq(`mod`(X, S1[k]), k = 1 .. nops(S1))

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

(4)

``


 

Download what_the_fook.mw

Hi, I realise it looks a little messy and I apologize for that, but I wanted to exclude the data pairs for which x or y are equal to 1, and the manner in which I have done has always worked for me thus far, so I would like to understand why or how I should produce output like this in the future thanks in advance.

 


 

`𝓃` := proc (X) options operator, arrow; numer(X) end proc:

`𝒹` := proc (X) options operator, arrow; denom(X) end proc:

delta := proc (x, y) options operator, arrow; piecewise(x = y, 1, x <> y, 0) end proc

[seq(seq(piecewise(`or`(`mod`(`&nscr;`(product(1-delta(`mod`(x, ithprime(j)), 0)/ithprime(j), j = 1 .. x)), y) = 0, `and`(`and`(`and`(`mod`(`&dscr;`(product(1-delta(`mod`(y, ithprime(j)), 0)/ithprime(j), j = 1 .. y)), x) = 0, igcd(x, y) = 1), x <> 1), y <> 1)), [x, y], NULL), x = 1 .. 20), y = 1 .. 20)]

[[1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1], [10, 1], [11, 1], [12, 1], [13, 1], [14, 1], [15, 1], [16, 1], [17, 1], [18, 1], [19, 1], [20, 1], [3, 2], [5, 2], [7, 2], [9, 2], [10, 2], [11, 2], [13, 2], [15, 2], [17, 2], [19, 2], [20, 2], [7, 3], [13, 3], [14, 3], [19, 3], [5, 4], [13, 4], [15, 4], [17, 4], [11, 5], [7, 6], [13, 6], [19, 6], [15, 8], [17, 8], [19, 9], [11, 10], [13, 12], [17, 16], [19, 18]]

(1)

``

``

``

``

``


 

Download 05062018.mw

For the most part, when I have come across bugs (or what appears to be bugs from my naive view, maybe it is left unevaluated for some means of diplomacy that I am not privy to) they are very difficult or impossible for me to find a way to get around.

But I have found alot recently that I was able to fix very quickly using functions that are already inbuilt in maple. Most have to do  with evaluating functions of trancendental arguements or composites of trancendental functions, but still, it leaves me feeling a little disturbed just how readily it was resolved.

I don't know exactly what to say, perhaps a 'dispatch' needed to be included in an inbuilt procedure to inform the 'engine' that a user has come across a bug of some kind at that address, so that maplesoft can keep the process of applying patches in house, avoiding having to enable the evolution/personal development the inept user that clearly stumbled upon the bug by sheer coincidence whist entering formula and observing output in a relatively sub human cognitive state?

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