# How to compute log919(2^82589933 − 1)?

i am not really sure how to compute this

using:

i can do base 2 and 10 but not 919…

any ideas?

Just change base of your log919.

log919(x) = log2(x) / log2(919) = log10(x) / log10(919) = ln(x) / ln(919)

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in theory this looks good, but how can i do it in practice?

I don’t know. 2^82589933 has around 2.5e7 digits. Without `-1` it’s fairy simple and can be done in `double` (using property of logarithm of power). Also, the difference between `log919(2^82589933 − 1)` and `log919(2^82589933)` is very, very small.

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first of all thx for helping me out! much appreciated!

still i’ll have to think about it for a bit

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I don’t know. 2^82589933 has around 2.5e7 digits. Without -1 it’s fairy simple and can be done in double (using property of logarithm of power). Also, the difference between log919(2^82589933 − 1) and log919(2^82589933) is very, very small.

i slept it over… but i still don’t get it

but using:

i can do it:

8389942.642…

but i am still very unhappy about the time it takes me to compute it! ( my first attempt to compute this took 4 minutes! )

now if i simply and naively enter the math into wolfram alpha:

log(2^82589933 − 1)/log(919)

it spits out the result pretty quick…

what am i missing here? any ideas?

Understanding when to use approximation is very important in this kind of calculation. Wolfram probably knows when an approximation will give the same answer, and uses that. `log(2^82589933 - 1)/log(919) - log(2^82589933)/log(919)` gives `0` in Wolfram Alpha. Wolfram gives the same answer as we get in Clojure, only with more digits:

``````(* (Math/log 2) 82589933 (/ (Math/log 919)))
;= 8389942.642742455
``````

I’m sure you can calculate the accuracy of the approximation if you need to. It’s probably accurate to far more than 100 digits.

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oh now i get it! ( kind of embarrassing that it would take me this long, after @generateme had already given me all that help! )

anyway, thank you very much for your explanation / patience! i really do appreciate it!

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I didn’t want to give you an actual solution

Just to emphasize why `-1` is numerically not important, imagine that you calculate previous power of 2, ie. `2^82589932`. The difference between `2^82589933` and `2^82589932` equals… `2^82589932` - which is huuuuge! Compare it with `-1`. And when you apply logarithm to and take the difference you get `0.1`. Such huge step (`2^82589933`) under logarithm, gives just `0.1`. So difference of `-1` is actually meaningless.

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yeah, now feels like things are starting to make a bit more sense

idk, for some reason i just didn’t see that i do not actually need to compute this largest of known prime numbers in the first place! … since… well since it is basically given as a power of 2, therefore i do already have the log2 of it in hand!

so:

log919(2^82589933 − 1) = ( for all practical intents and purposes ) log919(2^82589933) = log2(2^82589933) / log2(919) = 82589933 / log2(919) = 8389942.6427…

now seams to me that i can take away a couple of things from this:

• really good refresher on how log works
• explored some libs for dealing with huge numbers
• got reminded that some / often times i am a bit of a doofus
• got reminded that asking for help ( although slightly unpleasant / embarrassing ) is, more often than not, by far the smartest course of action!
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@taf out of curiosity, what motivated the use of base-919 logarithm? Seems uncommon

Btw: another frequently useful property of the natural logarithm is the approximation `ln(1+x) ~ x` for very small x. (From which you can deduce that `ln(A + B) ~ ln(A) + B/A` if `B << A`, which can be used to quantify @John_Shaffer’s point about approximation accuracy).

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what motivated the use of base-919 logarithm?

so i was doing a few of these advent of code puzzles… got a bit interested in the entire puzzle thing… and found the following very cool site:

so now i am doing a few of those as well… and that got me started thinking about prime numbers a bit… and more such math related things… wondering if perhaps i should try to come up with a little programming puzzle myself…

anyway… long story short… i ended up having a possibly cute little idea for how one could write a program for turning a large prime number into… well… kind of a pictorial ascii-art rendition thereof…

now… 919 is also prime and also a numeric palindrome… so it seamed to fit in well with the rest of my weird scheme

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