We often approximate the numbers for convenience and the digits to the rightmost side are neglected the most. I wanted to see of there is any pattern in the ignored right digits. For this problem, I have considered the increasing powers of a number. For example, have a look at this sequence.

2¹ = 2
2² = 4
2³ = 8
2⁴ = 16
2⁵ = 32
2⁶ = 64
2⁷ = 128

Have you found any pattern ?
The last digits are repeating as 2,4,8,6,2,4,8.
What if this is done not just for 2 but for the next numbers too ?
Let us see it first before plotting.

3¹ = 3
3² = 9
3³ = 27
3⁴ = 81
3⁵ = 243
3⁶ = 729
3⁷ = 2187

It is more apparent now isn’t it. The sequence is 3,9,7,1,3,9,7………

This being the case, I have taken it a few more steps ahead. I have run the same sequence with last two digits, three digits, four digits and so on. After proceeding to the higher numbers, I was taken aback by the visual patters that were emerging from the seemingly mundane and neglected string of integers.

For plotting, what has been done is Cⁿ for increasing n (which means increasing powers of a constant number)

2ⁿ for last two digits

2ⁿ for last three digits

2ⁿ for last four digits

I have skipped the other one digit numbers for convenience but there were similar patterns in them too. The most intriguing ones are shown below.

last 5 digits of  11ⁿ

last 6 digits of  11ⁿ

last 5 digits of  13ⁿ

last 6 digits of  14ⁿ

last 7 digits of 14ⁿ

last 6 digits of 16ⁿ

last 5 digits of 18ⁿ

last 6 digits of 18ⁿ

last 4 digits of 19ⁿ

last 5 digits of 19ⁿ

last 4 digits of 21ⁿ

last 5 digits of 21ⁿ

This post will be updated when other similar patterns are discovered.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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