683 is the sum of the five prime numbers from 127 to 149. Can you name those five prime numbers?

Like the number before it, 683 has a relationship with the number 11:

(2¹¹ + 1)/3 = 683. This relationship makes 683 the 4th Wagstaff Prime number. (Notice that 11 is the 4th odd prime number.)

2 raised to an odd prime number has produced many Wagstaff Prime numbers, but not always. For example (2²⁹ + 1)/3 is not a prime number.

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-16

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- 683 is a prime number.
- Prime factorization: 683 is prime.
- The exponent of prime number 683 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 683 has exactly 2 factors.
- Factors of 683: 1, 683
- Factor pairs: 683 = 1 x 683
- 683 has no square factors that allow its square root to be simplified. √683 ≈ 26.13427.

**How do we know that ****683**** is a prime number?** If 683 were not a prime number, then it would be divisible by at least one prime number less than or equal to √683 ≈ 26.1. Since 683 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 683 is a prime number.

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**A Logical Approach to solve a FIND THE FACTORS puzzle**: Find the column or row with two clues and find their common factor. (None of the factors are greater than 12.) Write the corresponding factors in the factor column (1st column) and factor row (top row). ** Because this is a level three puzzle**, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

I don’t think I’d heard of Wagstaff Primes before, which is odd since they seem to have gotten worthy of a name about the time I was in college and grad school.

There are so many different kinds of prime numbers, far more than I had heard of when I went through college, too. There will likely be other kinds named in our lifetimes, too.