For 'Higher' Mathematics: Behaviour of Prime Numbers.?

Question

Euler commented"Mathematics have tried in vain to this day to discover some order in the sequence of prime numbers, and we have reason to believe that it is a mystery into which the mind will never penetrate"(Havil 2003,p.163). In a lecture, D.Zagier commented,"there are two facts about the distribution of prime numbers of which I hope to convince you so overwhelmingly that they will be permanently engraved in your hearts. The first is that, despite their simple definition & role as the building blocks of the natural numbers, the prime numbers grow like weeds among the natural numbers, seeming to obey no other law than that of chance, & nobody can predict where the next one will sprout. The 2nd fact is even more astonishing, for it states just the opposite: that the prime numbers exhibit stunning regularity, that there are laws governing their behaviour, & that they obey these laws with almost military precision.
Please add your thoughts - serious thinkers of chance.

Answer 1

Much like the universe in general, prime numbers appear to obey several disparate "laws", yet at the same time, just like the universe in general, it currently seems beyond the scope of the human mind to come up with a "general theory of everything." Much about prime numbers is known, but much more remains unknown.

Remarkably, given any large natural number 'n', one can make a fairly accurate estimate of the cardinality of prime numbers less than 'n'. That is because analytic number theory has shown that as 'n' approaches infinity, the cardinality approaches n/ln (n).

Algebraic number theory has reached other remarkable results. Most significantly, one can prove that for any number a relatively prime to a prime p, we have a^(p-1) congruent to 1 mod p. This gives us a test by which we can conclude that a given number is composite, if the calculation gives any answer but 1 mod p.

There are many conjectures about primes that we do not know. A favorite of mine is the Twin Primes Conjecture, which speculates that there are infinitely many primes p for which p+2 is also prime. Although a paper offering a proof for this conjecture was written a few years ago, it was unfortunately retracted when it was discovered that an error was made in one of the lemmas.

Answer 2

One interesting tidbit regarding the regularity of prime numbers is that if you make a grid of squares and start writing the integers starting in the center with 1, spiraling outward, then the primes will tend to lie in diagonal patterns, and not be randomly distributed as you might initially think.

See the web link below for more

Answer 3

That quote is an interesting summary of prime numbers. Another thing that comes to mind is Goldbach's conjecture (every even integer > 2 is the sum of two primes). So simple to state, but nobody has ever come up with a proof.

Answer 4

I assume this has been tried by someone, somewhere in the world, but how about looking at prime numbers in their binary form to see if there is a pattern. Binary is yes or no and numbers are either prime or not prime so it seems reasonable to look at the prime numbers in binary.

Answer 5

The assessment is excellent, besides the shown fact that it somewhat is by utilising no capacity an equivalence. it somewhat is thrilling that this actual quote from Russel is rather contradictory. If Plato's Republic is something to bypass by utilising besides. And no, arithmetic is hardly 'complicated technological know-how'; it want no longer be sure by utilising the shakles of certainty. genuinely it helps the certainty of certainty, besides the shown fact that it could challenge into plenty extra. that's what Hardy became into somewhat attempting to assert approximately organic arithmetic; there is attractiveness in its uselessness (evaluate Kant on aesthetic judgement, disinterested liking, etc.) and certainty possibly in basic terms in technique (evaluate set-theoretic inadequecies, even Zermaelo Frankel concept with the axiom of determination could have inconsistencies, regardless how esoteric they may well be)! while utility is discovered, it in basic terms strikes on and subsequently is in endured evolution.