Oh come on, I am sure Eratosthenes had loads of papyrus to scribble on
i don't know if i have correctly spelled is name but lets call him Era
so Era was the guy who had the idea to find out the prime number by crossing all the multiple of the smallest number he had not crossed yet (exept one) till he has reach a certain number, assuming this smallest number was a prime number, reapeting the operation till all the numbered are crossed or marked as prime ( i supose you allready know the story )
the problem is that at the time he was leaving near the sea and a sandy beach and as theyre were no paper he had draw a you know what on the sand and he had to carry big and heavy stones to marks all those numbers
so here is the mathématical riddle:
when Era wants to mark the multiples of the nème primes number (lets say one is the first) and wants to do it for all the multiples <= M, how many stones does he need to carry considering this is no use to mark one place with more than one stone
this is to check my solution (i consider it wont give you any help whit the problem of checking if i'dd wrote my own solution )
oh yes certainly ill posed but i had a few excuse to do so
one i have difficulties to find such simple words as ... you know what!
two if i forgot to tell for exemple that Era had forgoten to count on the way back for n-1 prime the numbers of stones he had to put the next time, its because i just invented the story 1 hour before posting it! (i wanted to verify an assertion of mine which was wrong (in fact) but that would had made the problem much more easyer than it seem to be)
as a mater of fact i'm still interested in the procces
and i'll try to find the solution of this Riddle before anybody gives an anwser
i don't know if ancian greek had papyrus or paper to write on, espacialy if there were not rich and i believe paper (or papyrus ) was devoted to important spiritual or comercial maters, not for the elucubration of mathématician , i suppose you had to be a mathématician of a certain importance to be able to use paper at the time of Era (i'll check out this point one of these days (or an other))
it would be a pity if not because: i intended to build a riddle with some camels carrying on those stones and needing a dialectical more water per kilometers the more they are charged
so i supose they would had chalk or write on the sand or mediting a long time before writing in those time ()!
as i know there were no telescope or such thing on those time i cannot pretend they would had try to communicate the notion of prime number to the hinabitant of the moon!
Maybe this one Era proposing the following deal to a maharadja :I make the previously explained above little stuff using only square (n*n) grid and you give me the n diamonds if n is the side of the littlest square for wich i can found a prime number p for wich i have to put n new stone (the golden stone on p excluded) proccessing as explained.
may be it is ill posed, easy or difficult, but there is a reasonable motivation to justify the riddle!
the previous one was a little hard and the assertion that i'll give you the answer before any one give it don't seem to be very pretentious in regards of logical reasons
any way i 'll teel you why i'm interrested in the proccess
it's about the twin prime conjecture
i'm considering the two last twin primes Pm and P(m+1)
and considering the fact that Era would had allready donne the procces t'ill a number as big as it would need (he wil then needed some star treck tools not only camels) till P(m-1) on a gridd the whith would be the product of the m-1 first prime numbers
so it is easy to see that the Pm and the P(m+1) colums would be empty as this time of the process!
so i m searching a number M that when Era would had done is stuff over it (all the number unmarked then and < M*M would be prime ) then we would know that if we call P the the line of the grid for wich P*(product of th P(m-1)first prime + P(m+1)) < M*M then we would could tell that we would had put less than P/2 new stones on those perticular two collums
may be some one has explore this way uselessly maybe not!
personnaly it seems to me that it would be a reasonable statement pretend.
but i have no perticular knowledge (only presomption) on the statisticals propertys of the rèpartition of prime numbers
so if it's not useless i would say that somebody else than me would have more chance to give an answer before i do
and as many conjectures seems to be made to be difficult to be proved (or denied) easely it is probable that there is no hope following this track!