New Fastest Method to Find p & q of Given n.

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To find p and q of z
Where, z = 1384129

1st Step – Find required 6 ^ n = m.

Here as we can see the z = 1384129 is having a seven digits of integers.
Therefore, check above at ‘1.1 - Reduction value of z,’ it states
1000000 as seven digits z requires 6 ^n = m, where m can be up to 3 to 4 digits.
So, we will take two assumptions having 2 and 4 digit integer.
6 ^3 = 216 …… 3 digit integer.
6 ^4 = 1296 …….. 4 digit integer.

2nd Step – Dividing z by m to get r and calculation by guessing.

Note - Always start calculation by following the last 6 ^ n = m as first. We do this
in order to reduce the calculation timing.
By taking the second 6
n = m,
6 ^4 = 1296
z = 1384129, m = 1296 substitute it at below formula -
1384129 / 1296 = 1068.00 ( ignore the decimals)
r = 1068
Using Iterative Method For Getting Close Approximation’s -
We can start by guessing g as starting from 2,3, 4, 5, 6, 7, 8, 9, 10… and use it at
below formula to find final results.
Formula -
(1068 × 6 / 2 ) +5 = 3209
(1068 × 6 / 3 ) +5 = 2141
(1068 × 6 / 4 ) +5 = 1607
(1068 × 6 / 5 ) +5 = 1286.6
(1068 × 6 / 6 ) +5 = 1073
.
.
.
.
1068 × 6 / 13 ) +5 = 497.923..
Ignore the decimals and we get 497
Let s = 497
We get very close approximation final answer only at 13th step.
The close approximate difference s need to adjusted by subtracting as s - 1, s - 2;
s - 3, s - 4 ….. s - n, and adding as s + 1, s + 2; s + 3, s + 4 ….. s + n,
simultaneously, so on until we get the final answer.
Here below, we have just done subtraction operation for presentation. One must
do both side operations to figure out the final answer.
497 – 1 = 496
497 – 2 = 495
497 – 3 = 494
.
.
497 – 6 = 491
497 – 6 = 491, 1384129 is divisible 491
So, p = 491
Therefore, 2819 × 491 = 1384129


Check more examples solutions at paper - https://www.academia.edu/88229803