Picture a side view of prime steps where all the composites and the
one are the treads and the primes with their associated treads = x.
All empty squares (cells) above fill the horizontal rectangle.
The sum of these empty squares = y and is equal to each prime -1
EG. The first 6 primes = The sum of 6 primes =41 -6 = y =35.
The total height at this point = 6 representing the 6 primes.
The total length = number of integers <13 =12.

So now the height and length of this rectangle are known.
(6 X 12) = 72.
So 72 - y(35) = x = 37 where x is the number of units in the prime steps
ending with 13 and y = sum of the first 6 primes =41 -6 = 35 units in the
upper half of the empty cells above the steps forming the expanding rectangle.

Except for the first few prime steps there are more
cells in x then in y as pn --->oo.

100,000th prime = 1,299,709

Rectangle = 100,000 X 1,299,708 = 129,970,800,000
The Sum = 62,260,698,721 - 100,000 = y = 62,260,598,721

Rectangle size.
Hieght = 100,000
Length = 1,299,708
100,000 x 1,299,708 = 129,970,800,000

129,970,800,000 - y = (62,260,598,721) = x =(67,710,201,279)

x that represent the prime steps and treads and the (1) and composite treads
which = x and all the empty cells above in the rectangle = y.

An interesting layout of the primes.

Dan

On Friday, January 13, 2023 at 10:08:48 AM UTC-5, Dan joyce wrote:
> Picture a side view of prime steps where all the composites and the
> one are the treads and the primes with their associated treads = x.
> All empty squares (cells) above fill the horizontal rectangle.
> The sum of these empty squares = y and is equal to each prime -1
> EG. The first 6 primes = The sum of 6 primes =41 -6 = y =35.
> The total height at this point = 6 representing the 6 primes.
> The total length = number of integers <13 =12.
>
> So now the height and length of this rectangle are known.
> (6 X 12) = 72.
> So 72 - y(35) = x = 37 where x is the number of units in the prime steps
> ending with 13 and y = sum of the first 6 primes =41 -6 = 35 units in the
> upper half of the empty cells above the steps forming the expanding rectangle.
>
> Except for the first few prime steps there are more
> cells in x then in y as pn --->oo.
>
> 100,000th prime = 1,299,709
>
> Rectangle = 100,000 X 1,299,708 = 129,970,800,000
> The Sum = 62,260,698,721 - 100,000 = y = 62,260,598,721
>
> Rectangle size.
> Hieght = 100,000
> Length = 1,299,708
> 100,000 x 1,299,708 = 129,970,800,000
>
> 129,970,800,000 - y = (62,260,598,721) = x =(67,710,201,279)
>
> x that represent the prime steps and treads and the (1) and composite treads
> which = x and all the empty cells above in the rectangle = y.
>
> An interesting layout of the primes.
>
> Dan
Prime risers and corresponding prime treads that is.