So a gap of 5 occurs when we take 3*5*2=30, and subtract 3 and 5. Then when we subtract (rather than add) any of the odd numbers smaller than or equal to 1000, we must get a composite. To find 1000 consecutive composites, we can look for LCM(2,3.,999,1000) as Sir Col suggests, but a better way (finding a smaller number by a factor of about n/2) would be to generate LCM(3,5,7,9,11,13.,999) (LCM of all the odd numbers smaller than or equal to 1000), then multiply by 2. First, as towr noted, even numbers are always composite (we'll ignore 2), so we need only find odd gaps of consecutive composite integers. We can do a little better than that, perhaps explaining why the first gap of 5 occurs from 24 to 28. That's an interesting link, towr especially the table of gaps: If M n represents the smallest number that is evenly divisible by each of the integers 1 to n, then M n+1+2 represents the first number in a guaranteed chain of n composite numbers. On Oct 8 th, 2003, 3:08am, towr wrote: So I'd say your method is flawed. Impudens simia et macrologus profundus fabulae « Last Edit: Oct 8 th, 2003, 3:14am by towr » So the chains for 2n and 2n+1 allways start at the same place. The smallest chain for 4 non-primes not surprisingly starts at the same number, since there is always an odd number of numbers between two primes over 3. In the first 10,000 primes, the distance between adjacent primes is definately 5) Of course, the actual sequence may be very far up the numbers. It doesn't work for smaller numbers - for example, you can easily find 5 non-primes before 5!+2.6 Is TenaliRaman's series also the first natural series of 1000 consecutive non-primes? On Oct 8 th, 2003, 12:47am, towr wrote: So, the next question. that's probably better than my -1000 to -1Ĭhanged the original to require natural numbers. Wikipedia, Google, Mathworld, Integer sequence DB that's probably better than my -1000 to -1 Some people are average, some are just mean. Self discovery comes when a man measures himself against an obstacle - Antoine de Saint Exupery In general one can prove the existence of k consecutive non-primes, How about supercalifragilisticexpialidociouspuzzler There exist a series of 1000 consecutive natural numbers, non of which is prime.Īdded the contraint of "natural numbers" specifically to bypass Towr's negative number suggestion Topic: 1000 consecutive non-primes? (Read 5930 times) RIDDLES SITE WRITE MATH! Home Help Search Members Login RegisterĮasy (Moderators: Grimbal, towr, Icarus, william wu, SMQ, ThudnBlunder, Eigenray) « wu :: forums - 1000 consecutive non-primes? » Wu :: forums - 1000 consecutive non-primes?
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