One of the interesting things about the new method is that it doesn't use any kind of computer, either classical or quantum. Instead it involves a physical quantum system—a " quantum simulator "—that, when encoded with the number to factor, exhibits a probability distribution of energy values that is equivalent to the probability distribution of the prime factor candidates of the encoded number.

Every pair of primes that solve the problem re-arrange themselves to form a regular pattern: the band spectrum of the quantum simulator. The general idea behind the quantum simulator is something called the "factorization ensemble," which the researchers introduced previously. It is based on the idea that the primes are ordered from least to greatest for example, 2 is the first prime, 3 is the second prime, and is the 26 th prime. It's also possible to take the square root of any number, and then compare the result to the closest prime.

For example, the square root of 27 is a little more than 5, which is the third prime. By the definition of a factorization ensemble, this means that 27 belongs to the factorization ensemble of 3. The physicists then showed that they could transform the factorization ensemble function into a function from quantum physics the inverted harmonic-oscillator function. After many more steps, they eventually showed that the predicted energy spectrum of a quantum system corresponds to the distribution of primes in the factorization ensemble of a number.

From this information, the researchers can determine the probability that a prime is a factor of that number. To test the validity of their method, the physicists tested certain numbers and compared their results to the actual distributions obtained using prime number tables, and found very similar distributions.

The physicists theoretically demonstrated that the proposed quantum simulator can factor numbers that are many orders of magnitude larger than those that have been factored with quantum computers. In their paper, they report the results of using their method to determine the probability distribution of the prime factors of a number with 24 digits. Further, the method does this with far fewer resources than required by classical factoring algorithms. One final point of interest is that the new method has strong connections to the Riemann hypothesis, which, if true, would suggest that the prime numbers are distributed in a predictable way—in the same way as the distribution of the zeros of the Riemann-zeta function.

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Proving or disproving the Riemann hypothesis is one of the greatest unsolved problems in mathematics, and one of the Clay Mathematics Institute's Millennium Prize Problems. Going forward, the researchers are currently working on the experimental implementation of the quantum simulator by using two entangled particles in a Penning trap. The aim is to build a quantum factoring simulator experimentally. If successfully implemented, numbers many orders of magnitude bigger than those available for its quantum processing using Shor's algorithm will be factorized and, as a by-product, the Hilbert-Polya conjecture will be tested experimentally.

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## Project Euler 47: Find the first four consecutive integers to have four distinct primes factors.

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Learn more Your name Note Your email address is used only to let the recipient know who sent the email. Neither your address nor the recipient's address will be used for any other purpose. As such there are two possible interfaces for calculating the factors of an integer.

This code shows both styles of implementation. Note that much of the code here is ceremony put in place to have this be something which can actually compile. The function form is implemented in terms of the predicate form rather than duplicating all of the predicate code.

## Highly composite number | Large Numbers Wiki | FANDOM powered by Wikia

This makes the code more immediately understandable. This implementation makes use of Mercury's "state variable notation" to keep a pair of variables for accumulation, thus allowing the implementation to be tail recursive. One of them is an "in"-moded variable and the other is an "out"-moded variable. Accumulator is the "in" portion in the ensuing code.

### Factoring Worksheet Generator

Using the state variable notation avoids having to keep track of strings of variables unified in the code named things like Acc0, Acc1, Acc2, Acc3, etc. There is a builtin factors n , which takes an optional second parameter to include 1 and n, so eg? This one uses a range of integers up to the target number and just filters it using the Where-Object cmdlet. It's very slow though, so it is not very usable for larger numbers. Here the range of integers is only taken up to the square root of the number, the same filtering applies.

Afterwards the corresponding larger factors are calculated and sent down the pipeline along with the small ones found earlier. Much better realize that factors come in pairs, the smaller of which is no bigger than sqrt n :. This REXX version also supports negative integers and zero. It also displays a final count of the number of primes found.

This should work in all Bourne-compatible shells, assuming the system has both sort and at least one of bc or dc. Combining these ideas, losing the dummy variable, and cleaning up some other cruft, we have. Create account Log in. Toggle navigation. Page Discussion Edit History. Factors of an integer From Rosetta Code. Jump to: navigation , search.

Works with : as version Raspberry Pi. Works with : QBasic.

Works with : Fortran version 90 and later. Library: Icon Programming Library. It's a standard sieve of Erasthones. I can get to about 9e7 in 25 seconds on my Linux machine. There are much more clever ways to do this using sieving. I thought a simple brute force way would be a start. This is basically Project Euler problem , by the way.

Spawns goroutines to count sections of the number space. This entry does not compare to several of the other ones already up, but I wanted to post my answer since I had fun working on this. That's about all there is to it. My code iterates from 3 on by twos until it detects that it has hit or exceeded the time limit, at which point it spits out the answer.

Running on a different system Windows 8. Simple minded sieve of Eratosthenes, vectorised as much as possible. R's not really designed for this sort of problem and I'm pretty sure it can be made faster. Using an incremental Sieve of Erathostenes.

### Example worksheets

Some care does need to be taken that a marked value is marked with its lowest divisor, but the implementation is otherwise fairly straight forward. Surely Julia can do better but this is what I have for now. This does 5e7 in about 60 seconds on JuliaBox but I can't test locally yet.

Hopefully by then I'll have thought of a more clever approach. Here we're creating a function lpf that iterates through sequential primes and checks the input for divisibility by each. The function returns the first divisor encountered, thereby obtaining the least prime factor.