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# beauty and the beast

## Question

consider the golden ratio.

1.618....

now, imagine multiplying this by each integer. keep the numbers it truncates to.

1

1.618*2 = 3

1.618*3 = 4

1.618*4 = 6

1.618*5 = 8

1.618*6 = 9

and so on.

its a very interesting sequence with some "nice" properties.

now here's my question. if you're allowed any irrational number;

which one goes through the most primes and fewest composites?

## Recommended Posts

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Prime numbers (greater than 3) are of the form 6n+1 or 6n-1

Since approximating 6n+1 using a rational number will be difficult as n increases, lets consider only 6n-1 cases.

As we are rounding down the numbers, an expression close to but only just less than 6 that gives all the numbers of form 6n-1, will have almost a third of the numbers prime.

Therefore using 5.99999999... we would get all 6n-1 numbers and nearly 1/3 of these numbers would be prime

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^ 5.999999... (continue forever) is actually 6.

And if it doesn't continue forever then at some point you will get 6n-2...

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