checking very large number for primality.

[a1call]

Never seen that before
Somehow it does make sense
I can't get a number higher than 24, no matter what number I add?
https://www.timecalculator.net/hours...now-calculator

[VBCurtis]

Originally Posted by a1call

How would that argument not apply to the following comment by yourself earlier( not to mention with conventional computing)? Hmmm?

If you knew anything at all about trial division or other methods to find small factors, you wouldn't be asking this question. Never mind algebraic factors- you know what those are, yes?
Try doing some reading- really. Your ignorance is on full display in this thread.

[JeppeSN]

Originally Posted by WhoCares

n= 39639600000000033079722130334105193516374462454515382070790605358914985114125041349652662917631491895468078068144227588488256698643731392656301687065828647039364869335097585180284485610320873304664240820377108832285808799206086668871981475355607012901091630830301025471386040357260999007608336089976844194508766126364538537876281839232550446576248759510420112471055243135957657955673172345352299040688058220310949388025140588819053919947072444591465431690373800860072775388686735031425736023817399933840555739331789612967251075090969235858418789282170029771749917300694674164737016209063843863711544823023486602712537214687396625868342705921270261329804829639431028779358253390671518359245782335428382401587826662256037049288785974197816738339397949057227919285478001984783327820046311610982467747270922924247436321534899106847502480979159775057889513728084684088653655309295401918623883559378101223949718822361892160105855110817069136619252398279854449222626529937148527952365200132318888521336420774065497849818061528283162421435659940456500165398610651670525967581872312272576910353953026794574925570625206748263314588157459477340390340721137942441283493218656963281508435329143235196824346675487925901422428051604366523321204101885544161429043996030433344359907376778035064505458154151505127356930201786304995038041680449884220972543830631822692689381409196162752232881243797552100562355276215679788289778365861726761495203440291101554746940125702944095269599735362222957327158451869004300363876943433675157128680119087

But where does that exponent come from? Why do you think it is particularly interesting? /JeppeSN

[science_man_88]

Originally Posted by JeppeSN

But where does that exponent come from? Why do you think it is particularly interesting? /JeppeSN

I tried to turn it into k*b^n+/-1 form but it didn't really help the k value was still way too large.

[LaurV]

Originally Posted by VBCurtis

Finding a factor to show compositeness is OP's only hope.

Subscribing.

What do some people are smoking these days? speed of light? trillions of operations per second? do you mean like... let's see... kilo..., mega..., giga... tera...

Tera-flops?

GIMPS can do a hundred of them and it is not optic.

And speed of light... what the hack is that? Are you talking about that thing which I can nowadays measure extremely accurate with a 500 dollars (cheap) oscilloscope, using one LED, two photoresistors, a mirror placed at 30 centimeters away, and a caliper?*** (the caliper is to measure the distance to the mirror )

That is, by far, not "instant". You need better then that... Neither quantum computers can help you here...

I think some people do not realize the magnitude of the numbers involved. Like it was already explained, by Curtis and others, GIMPS, with its hundreds of teraflops, can only test numbers with 7-8 digits at the exponent*. And these numbers have a very special form that makes them extremely easy to test for primality. Talking about random picks, we are not even able to test a number with an exponent of 6 digits** for primality. With ALL our planetary resources put together. See for example the quest to "find the smallest prime number with exactly a million digits".

Imagine that all atomic particles that we have a name for (electrons, protons, etc), from the visible universe come up to a number with 2 digits at the exponent (yes, they are less than 10^100 all together).

Then imagine that we (humanity) were not able in over half of a century to prove (or disprove) the Catalan conjecture, i.e. test if \(2^{2^{127}}-1\) is prime. Or, how somebody would call it, MMMMM2...

It goes like that:
\(M2\) is \(2^2-1=3\), prime;
\(M3=MM2\) is \(2^{M2}-1=2^3-1=7\) prime;
\(M7=MM3=MMM2\) is \(2^{M3}-1=2^7-1=127\) prime;
\(M127=MM7=MMM3=MMMM2\) is \(2^{M7}-1=2^{127}-1=17014...5727\) (39 digits) prime;
So, is the next one, \(MM127=2^{39\ digits\ exponent}-1\) prime?

Here we only have a 39 digits exponent, and the number has a very special form which is extremely easy to test for primality.

And we (humanity) were not able to do it in a hundred years.

And you want to test a number with over a thousand digits in exponent?

C'mon...

You will need either some "super-hyper-quantum" stuff, or a lot of new mathematical break-through...

---------------
(*) (no, not 8-9, that is when the base is 2, we are talking about base 10)
(**) (again, no, a number with a million digits does not have an exponent of 7 digits, because 10^one million is a number with one-million-and-one digits, i.e. 10 times bigger; 10^999999+x for a small x is the prime we are looking for)
(***) the light travels ~300000 km in a second, which would make a 300000 meters in a millisecond, equivalent with 300 meters in a microsecond, or 30 centimeters in a nanosecond. For an average oscilloscope with 0.5 nanoseconds time base, i.e. 0.1 ns per screen division, you will see a measurable delay between the direct light pulse and the pulse reflected in the mirror. You still need to rotate the mirror, and a light guide to switch which light goes to one of the two photoresistors, and which to the other, and measure in both ways, to eliminate circuit-related delays, and you need to do few measurements with the mirror at intermediary distances, for interpolation purposes, but at the end you will get an accuracy better than world-top scientists and scientific labs were able to do 60 years ago.

[Dr Sardonicus]

Originally Posted by VBCurtis

Finding a factor to show compositeness is OP's only hope.

I agree. It is conceivable that this could be done by finding a polynomial f(x) in Z[x] with an algebraic factorization in Z[x], for which f(10) = 10^n + 7 and neither algebraic factor evaluates to 1 or -1. Alas, I don't know any obvious candidates.

[a1call]

There has been a lot of nonsense uttered in this thread by the "expert" members and I do not really like to resort to authoritative sources.
However, here is a quote from an organization which probably knows a thing or two about computers and again very likely has a PHD or two on their payroll.

Optical data processing can perform several operations simultaneously (in parallel) much faster and easier than electronics. This "parallelism" when associated with fast switching speeds would result in staggering computational power. For example, a calculation that might take a conventional electronic computer more than eleven years to complete could be performed by an optical computer in a single hour.

https://science.nasa.gov/science-new...00/ast28apr_1m

The implication here that somehow conventional computers can perform factoring of large integers where Optical computers cannot is baseless nonsense, also.
I'm just saying.

[ET_]

Originally Posted by a1call

There has been a lot of nonsense uttered in this thread by the "expert" members and I do not really like to resort to authoritative sources.
However, here is a quote from an organization which probably knows a thing or two about computers and again very likely has a PHD or two on their payroll.



https://science.nasa.gov/science-new...00/ast28apr_1m

The implication here that somehow conventional computers can perform factoring of large integers where Optical computers cannot is baseless nonsense, also.
I'm just saying.

You are still not paying attention to the numbers implied to the calculus you proposed.

11 years vs 1 hour means a speedup factor of 100,000x while your requested check needs a time that probably surpasses 200x the actual age of the universe.

Even applying a reduction of a factor 100,000 using hypothetical "optical computers", we need more than 27,6 million years to perform it.

Not to mention tat the Universe has not enough atoms to store the digit for the computation...


Can you see now why everyone is laughing at you?

[retina]

Originally Posted by ET_

You are still not paying attention to the numbers implied to the calculus you proposed.

11 years vs 1 hour means a speedup factor of 100,000x while your requested check needs a time that probably surpasses 200x the actual age of the universe.

Even applying a reduction of a factor 100,000 using hypothetical "optical computers", we need more than 27,6 million years to perform it.

Can you see now why everyone is laughing at you?

Hey don't break the spell. Computers are magical devices, and optical computers are clearly doubly magical. People bringing in plain and simple facts into the discussion would threaten the fantasy world that some people maintain about how wonderful future computers will be. So even numbers that are impossible to fully expand and store within the currently known universe will magically become doable with optics. And most important is the colour yellow for some reason that I have yet to understand.

[danaj]

Surely quantum computing would be better than mere optical computing.

But more seriously, post-quantum methods using light-plane superpositions seems like the right solution for this. There's some good research in phase modulation of the planes as well, which shows a lot of promise.

[xilman]

Originally Posted by ET_

Can you see now why everyone is laughing at you?

Not everyone is laughing. There appears to be at least two groups who are not. Those who do not understand the mathematics behind estimates based on established technology, and those who are now giving serious thoughts as to how the mathematics or technology or both can be improved to the point where the stated task becomes feasible.

I'm in the second category but I very much doubt that I will be able to make useful progress in either field of endeavour.