Someone asked an interesting question on MathOverflow: given an odd number, can you always flip a bit in its binary…
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How probable is a probable prime?
A probable prime is a number that passes a test that all primes pass and that most composite numbers fail.…
Continue ReadingRelatively prime determinants
Suppose you fill two n×n matrices with random integers. What is the probability that the determinants of the two matrices…
Continue ReadingPrime plus power of 2
A new article [1] looks at the problem of determining the proportion of odd numbers that can be written as…
Continue ReadingEstimating the proportion of smooth numbers
A number is said to be “smooth” if all its prime factors are small. To make this precise, a number…
Continue ReadingPrime numbers and Goldbach’s conjecture visualization.
isn’t it?The German mathematician G. F. B. Riemann (1826–1866) observed that the distribution of prime numbers is very closely related…
Continue ReadingShare The π: Honoring Neglected Mathematical Constants
Subsequent occurrences are fairly common. Brun’s and Meissel-Mertens ConstantsPrime numbers make an appearance in two fascinating constants we’ll discuss next.…
Continue ReadingPython Pro Tip: Use Itertools, Generators, and Generator Expressions
Think of the memory such a list would occupy. It would be great if we had something that could just…
Continue ReadingStrong primes
There are a couple different definitions of a strong prime. In number theory, a strong prime is one that is…
Continue ReadingGoldilocks and the three multiplications
Mike Hamburg designed an elliptic curve for use in cryptography he calls Ed448-Goldilocks. The prefix Ed refers to the fact…
Continue ReadingTricks for arithmetic modulo NIST primes
The US National Institute of Standards and Technology (NIST) originally recommended 15 elliptic curves for use in elliptic curve cryptography…
Continue ReadingElliptic curve P-384
Possibly, but they have reasons to recommend methods that they believe foreign governments cannot break. The equation of the P-384…
Continue ReadingNew prime record: 51st Mersenne prime discovered
The largest known prime is nowWritten in hexadecimal the newly discovered prime isFor decades the largest known prime has been…
Continue ReadingCode doodling: isPrime
Then I’ll run the code on my laptop and time the results for each algorithm testing for primes in the…
Continue ReadingPrime numbers in complex domains are actually quite simple
Likewise, a number in Z[√−2] is said to be divisible by another number in Z[√−2] if the result of the…
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