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There are a lot of unsolved problems related to prime numbers.
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The prime numbers have always fascinated mathematicians. List of prime numbers less than one million.Ī natural number is called a “ prime number” if it is only divisible by $1$ and itself.įor example, $2, 3, 5, 7$ are prime numbers, although the numbers $4,6,9$ are not.What is the largest prime number less than one million.With these properties you drop your trials to a few hundred. Three-digit sums of $3$, $11$, and $27$ fail because the appended units digit is $5$.Ī three-digit sum of $7$ allows two candidates with $1$ or $9$ as a units digit, but other surviving three-digit sums give only one candidate per three-digit seed. Three-digit sums of $17$, $19$, and $21$ fail because we can't get the four-digit sum to $32$. Those where the sum of the three digits is even are rejected because the units digit would have to be even, which is a problem. You do not really need to keep all the three-digit starting numbers. Each four-digit number is then tested for primality. You append a units digit to each one to make the total digital sum $8$, $16$, or $32$ (not $24$ as this would imply a multiple of $3$). Then you can check the digitsums of the resulting primes directly.Īnother approach is to start with all the three-digit numbers from $100$ to $999$. The size of your team would need to be large enough to be able to perform all the primality-check calculations in the time allotted.Īnother approach (assuming you had a large enough team) is to divide the $1000-10000$ range up into chunks and have each person enumerate their section of the range and then use the segmented Sieve of Eratosthenes approach using the primes under $100$ to cross out the composite numbers. I assume you can at least use a calculator for this step, because otherwise this is probably too tedious a problem to do by hand. If they all fail the test, then the candidate number is prime. Then you test each one for divisibility into your candidate number. One way you can do this is by listing the (relevant) primes under $\sqrt = 100$: Then for each unique permutation of each case that falls within your acceptable range (also ending in $1, 3, 7$, or $9$), test if the number is actually prime. This leaves you with the following cases: Then you can eliminate the cases that can't possibly be rearranged to form a prime number (such as those missing a $1, 3, 7$, or $9$ to form the last digit of the prime). However, you already know the $24$ digitsum is a waste of time, since any number whose digitsum is divisible by $24$ will also be divisible by $3$, which means the number itself will be divisible by $3$ and therefore not prime.
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If you were working in a team and had to do this by hand, you could probably divide-and-conquer and have each person tackle a different digitsum in order to enumerate all the cases more quickly. How many ways can you write each of these as a sum of $4$ digits where each digit falls within $0 \leq d \leq 9$?
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The possible digitsums under that barrier that are divisible by $8$ (ignoring $0$) are $8, 16, 24, 32$. You know the maximum possible digit sum (ignoring the prime condition for a moment) is $36$, due to the digitsum from $9999$ within your range.