Prime Curios!: 683

683
(another Prime Pages' Curiosity)

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If we place 11 in the formula 2^p-1, we get the composite Mersenne number 2047, but if we place 11 in the formula (2^p+1)/3 [p is an odd prime], we get the prime number 683.

683 = prime(6*8*3) - 6*8*3. Note that the prime 683 is the smallest number greater than one with this property. Note that the next such number is also prime. [Firoozbakht]

The smallest number with complexity of height 5: 683 -¹-> 22 -²-> 10 -³-> 7 -⁴-> 6 -⁵-> 5. [Beedassy]

Regarding light, the maximum possible efficacy is 683 lumens per watt which corresponds to an efficiency of 100%. [Homewood]

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