AboutVijilant Expertise Most questions on number theory, divisibility, primes, Euclidean algorithm, Fermat`s theorem, Wilson`s theorem, factorisation, euclidean algorithm, diophantine equations, Chinese remainder theorem, group theory, congruences, continued fractions.
Experience Teacher of math for 50 years
Organizations ATL
Publications Journal of mathematics and its applications
Education/Credentials BSc Hons Liverpool
Awards and Honors State Scholarship 1955
Past/Present Clients I taught John Birt, former Director of the BBC in 1961. His homework book was the most perfect I have ever marked. And also the most neat. I could tell he was destined for great things.
One of my classmates was the poet Roger McGough, and I have a mention in his autobiography.
Question Hello, I was reading "Archimedes' Revenge" which states that a simple formula that generates all prime numbers has yet to be found. It also shows "Ulam's doodle" which shows that prime numbers when arranged in a spiral will yield diagonal patterns. Just for fun I set out to see if I could find my own pattern in primes and I soon "discovered" (it's probably well known)a pattern that unlike Ulam's, seems to recur flawlessly. It seems to me that once you know of a definite pattern, a simple formula for generating prime numbers without trial and error testing is possible. The problem is, I don't have a clue how write any formula more complex then n + n + 1 = (an odd number). Say for an (very simple and hypothetical) example, I found that a certain number (I'll call Craig's numbers) occurs once every 2 numbers, then 4, then every 8 numbers (like 5, 7, 11, 19). How would I then take the 2 4 8 pattern I found and apply it via a formula to find a much larger number? I guess what I'm asking for is direction in taking patterns like these and converting them into formulas without being told to "go back to school". I'm also curious as to whether there has been any progress towards finding such a formula since Archimedes' Revenge book was published. I'm sorry I can't elaborate further on the pattern itself. Thanks in advance for any help you can offer.
Answer Hello Craig
I'm afraid there is still no simple formula for generating all prime numbers, nor is there a quick way to work out a formula. You just have to learn the different types. However, it is possible to use a database of sequences like http://www.research.att.com/~njas/sequences/
Try typing the sequence 5,7,11,19,35,67,131 in. It will give you the answer 2^n + 3.
Ulam's spiral is interesting. The author A.C. Clarke may have given him the idea.
Remarkably the noted author Arthur C Clarke described this spiral in his book City and the Stars (1956, Ch. 6, p. 54). Clarke wrote, "Jeserac sat motionless within a whirlpool of numbers. The first thousand primes.... Jeserac was no mathematician, though sometimes he liked to believe he was. All he could do was to search among the infinite array of primes for special relationships and rules which more talented men might incorporate in general laws. He could find how numbers behaved, but he could not explain why. It was his pleasure to hack his way through the arithmetical jungle, and sometimes he discovered wonders that more skillful explorers had missed. He set up the matrix of all possible integers, and started his computer stringing the primes across its surface as beads might be arranged at the intersections of a mesh."
Clarke never actually performed this forward thinking experiment, leaving discovery of the unexpected properties of the prime spiral to Dr. Ulam seven years later.
