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Current Events > What is the smallest integer N such that Phi(N)/N > .999 ?

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[deleted]
10/20/23 2:02:48 PM
#4:


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[deleted]
10/20/23 3:03:11 PM
#10:


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[deleted]
10/20/23 3:03:11 PM
#12:


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FLOUR
10/20/23 4:53:39 PM
#1:


Here Phi represents Euler's totient function, which counts the total number of integers less than x that are relative prime to x. For example, Phi(10) = 4 since 1,3,7,9 are relatively prime to 10. In this case, Phi(10)/10 equals 2/5 which is way less than .999 and isn't a candidate. Also, this does not include N=1, which is an ambiguous case.

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dummy420
10/20/23 4:54:28 PM
#2:


Are you trying to summon Satan or something?

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MatzoTov
10/20/23 4:54:30 PM
#3:


The smallest integer N such that (N)/N > 0.999 is N = 3.

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Frostshock
10/20/23 5:02:33 PM
#5:


1 obviously.

Other than that, the smallest prime number above 1000.

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LSGW_Zephyra
10/20/23 5:03:08 PM
#6:


Is this one of those 1 = .99999~ things?

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Paragon21XX
10/20/23 5:05:13 PM
#7:


The phi(N) of a prime number is always N-1, so the smallest prime where phi(N)/N>0.999 is 1009.

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BewmHedshot
10/20/23 5:08:54 PM
#8:


The density of primes is asymptotically proportional to 1 / ln(N) so the crossover point is probably somewhere around e^1000 or roughly 10^434.

If there is some number lower than that which simply happens to exist in a particularly prime-sparse portion of the number line you'd likely have to find it via brute force.

Oh relative prime I misread oh well.
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FLOUR
10/20/23 5:09:35 PM
#9:


Yup, it's 1009. Now for what values of N is Phi(N)/N = .5 ?

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Robot2600
10/20/23 5:21:39 PM
#11:


1 is not prime. do we count it anyway for the phi function?

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Robot2600
10/20/23 5:23:31 PM
#13:


also, ChatGPT will solve this for you if that's all this is about

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Nota214
10/20/23 5:24:23 PM
#14:


The smallest integer N is n. Just look at it. It's right there in the word integer.
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Robot2600
10/20/23 5:25:30 PM
#15:


cause it seems like literally nothing will work. You can't have more primes inside a series of integers than the N, like ever.

so 2, i guess is the answer.

phi(2) = 2 [1, 2]

2/2 = 1

1 > .999

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Robot2600
10/20/23 5:25:59 PM
#16:


3 also works and then nothing else works

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Tyranthraxus
10/20/23 5:26:36 PM
#17:


Robot2600 posted...
1 is not prime. do we count it anyway for the phi function?
Question just asked about integer not prime.

1 works

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Robot2600
10/20/23 5:28:02 PM
#18:


Tyranthraxus posted...
Question just asked about integer not prime.

1 works

he said "Also, this does not include N=1, which is an ambiguous case."

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LSGW_Zephyra
10/20/23 7:26:36 PM
#19:


Math nerds are cute

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lderivedx
10/20/23 8:04:31 PM
#20:


FLOUR posted...
Yup, it's 1009. Now for what values of N is Phi(N)/N = .5 ?

2^k for k \geq 1 and no others.

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Robot2600
10/23/23 8:39:26 AM
#21:


not even a ty for doing this guy's hw smh

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