June 2020

[tgan]

http://www.research.ibm.com/haifa/po.../June2020.html
Puzzle-Master Oded is leaving

[uau]

Has anyone been able to solve this? I wrote a program to find all numbers with the given number of relative primes, but found no exact match for the divisor sum among those. Closest I got to the target sum in absolute value was this:

sage: x=7943551318529981932577436079331984148138176
sage: factor(x)
2^6 * 7 * 13 * 29 * 1223 * 144323 * 586543 * 7695239 * 59035244685044657
sage: euler_phi(x)
3031634148236289733373855928919180891127808
sage: sum(divisors(x))-x
12142617410592093155511288408870474224367424

So 1214261... instead of 1214268...

By the way the phrasing of the question talks about "natural numbers smaller than 278" etc, but seems to exclude 0 - this would be ambiguous without the example values.

[a1call]

Originally Posted by uau

By the way the phrasing of the question talks about "natural numbers smaller than 278" etc, but seems to exclude 0 - this would be ambiguous without the example values.

Do you mean include 0?

Pari-gp code:

counter=0
for(i=2,278,{
    if(gcd(278,i)<2,counter=counter+1);
})
counter

Output is 137 not 138.
1 is not a prime number (at this point in history) and thus not coprime to any number. and 0 is not coprime to any number:
gcd(278,0) = 278

It looks like the Puzzle-Master is considering 1 as coprime to the solution.

I fail to see how 0 is being relevant.

ETA:

The numbers 1 and −1 are the only integers coprime to every integer, and they are the only integers that are coprime with 0.

https://en.wikipedia.org/wiki/Coprime_integers

So 1 is a coprime by convention/definition.
But 0 is still irrelevant as far as I can see.
corrections are appreciated.

[uau]

Originally Posted by a1call

1 is not a prime number (at this point in history) and thus not coprime to any number.

Not being a prime doesn't stop it from being coprime to something else. gcd(x, 1)==1, so 1 is coprime to x, whatever x is.


But yeah 0 is not relevant, I wasn't really thinking when writing that...

[a1call]

We cross posted. I stand corrected on 1. But, I still think it's a matter of convention and not logic.
ETA: In the same way 1 was considered to be a prime number at some point until someone with authority decided it was not. And Retina has his own logic.
ETA II: 1 is the same number it has always been.
ETA III: Needless to say that I still consider Pluto to be a planet. It's the same heavenly body it was before someone decided to demote it.

[LaurV]

Originally Posted by a1call

Pari-gp code:

counter=0
for(i=2,278,{
    if(gcd(278,i)<2,counter=counter+1);
})
counter

A lot of work you did there, man...
gcd, if, for,... grrrr
(edit: even if not talking about the futile accolade)
haha

gp>eulerphi(278)
%1=138

[a1call]

Thank you LaurV, But I actually new about that function.
One day in elementary school my geometry teacher asked if anyone could prove some particular theorem. I raised my hand described a valid proof which took me about 20 minutes to describe in the 45 minutes class. At this point the teacher replied, that is correct but it is equivalent to turning your arm behind your head and then taking the spoon to your mouth when eating. Afterwards he proved the theorem with 3 sentences.
We grow up on the outside but we are the same person we were when we were kids.
Kind of like Pluto.

[a1call]

Originally Posted by uau

Not being a prime doesn't stop it from being coprime to something else. gcd(x, 1)==1, so 1 is coprime to x, whatever x is.

You are correct of course, but only iff, (generic)-you decide to exclude 1 from the definition of one or the other by some twisted logic but not both.

Google >> define coprime numbers >> Click-Top-Result >>

When two numbers have no common factors other than 1.

https://www.mathsisfun.com/definitions/coprime.html

Google >> define prime numbers >> Click-Top-Result >>

A whole number greater than 1 that can not be made by multiplying other whole numbers.

https://www.mathsisfun.com/definitio...me-number.html

[uau]

Originally Posted by uau

but found no exact match for the divisor sum among those

Was due to a bug causing the program to skip some values.

[Dieter]

Originally Posted by uau

Was due to a bug causing the program to skip some values.

Big numbers, not many solvers.

Can anyone check the following - it‘s not the solution, of course:

x = 2**7 * 5 * 23 * 127 * 659 * 53323 * 1876187 * 97544836889 * 665320793909
= 7998766649128898059663516612687535453720960

euler_phi(x) = the wanted value

sum(divisors(x)) - x = 12142697391577851168337274092012830083559040

I would like to know, if at least one prime of the solution > 2**64...😉

[bsquared]

Originally Posted by Dieter

Big numbers, not many solvers.

Can anyone check the following - it‘s not the solution, of course:

x = 2**7 * 5 * 23 * 127 * 659 * 53323 * 1876187 * 97544836889 * 665320793909
= 7998766649128898059663516612687535453720960

euler_phi(x) = the wanted value

sum(divisors(x)) - x = 12142697391577851168337274092012830083559040

I would like to know, if at least one prime of the solution > 2**64...😉

>> x=2^7* 5 * 23 * 127 * 659 * 53323 * 1876187 * 97544836889 * 665320793909
7998766649128898059663516612687535453720960
>> totient(x)
3031634148236289733373855928919180891127808
>> sigma(x,1)-x
12142697391577851168337274092012830083559040
>>