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June 11th, 2018, 01:15 AM  #1 
Newbie Joined: Jun 2018 From: Groningen Posts: 3 Thanks: 0  Finding regularity within exponents
Hello everyone, I love patterns and I try to find regularity in many things. So I have this habit of playing with numbers and finding regularities (but I'm no mathematician). Yesterday, I tried to find a regularity within exponents. I managed to find it. However, the outcome was quite weird but I think it is right. I wrote a blogpost on my webpage to explain what I did. Bear in mind, I'm no mathematician so my writing might be a little bit confusing, but I guess that the table explains most of the process. And because of the fact that I'm no mathematician, I would love to hear any feedback on this. Whether it makes sense, has any meaning or whether this is useless. Here is the link to my blogpost Finding a linear pattern in exponents � JUSTIN TIMMER Last edited by skipjack; June 11th, 2018 at 06:30 AM. 
June 11th, 2018, 08:13 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 19,042 Thanks: 1618 
You made a slip near the beginning, as 2^5 = 32, not 36. A number written as 2^x is called a power of x, especially when x is a whole number. For the numbers a^x, a^(x+1), a^(x+2), where a > 0, the (first) differences are a^(x+1)  a^x and a^(x+2)  a^(x+1). If a is not 1, (a^(x+2)  a^(x+1))/(a^(x+1)  a^x) = a. Differences of first differences are called second differences, etc. The numbers 1, 2, 6, 24, 120, 720, etc., can be written as n! for the nth such number. Thus, 1! = 1, 2! = 2, 3! = 6, etc. By convention, 0! = 1. Note that n! may be read as n factorial. Factorials have lots of interesting properties. Last edited by skipjack; June 11th, 2018 at 11:52 PM. 
June 11th, 2018, 05:33 PM  #3 
Senior Member Joined: Jul 2012 From: DFW Area Posts: 621 Thanks: 86 Math Focus: Electrical Engineering Applications 
Another small mistake: Right before Table 1, the "sixth time difference a constant of 120 remained". 120 should be 720 here, of course, as you later state. Since you like detecting patterns, what happens to the constant differences (the factorials as pointed out above) if instead of 1,2,3,4, etc. (an input difference of 1), you use 1,3,5,7,etc. (an input difference of 2); or 1,4,7,10,etc. (an input difference of 3)? Edit: Hint  Divide by the factorial value. Last edited by jks; June 11th, 2018 at 05:38 PM. 
June 13th, 2018, 02:25 AM  #4 
Newbie Joined: Jun 2018 From: Groningen Posts: 3 Thanks: 0  Thanks
Thanks for your responses and your feedback on the mistakes! I changed them now on my blog. Great to learn about the factorials. I am wondering about the interesting facts about factorials! That opens the doors to a deeper world for me. Jks, I am not really sure if you meant this, but I took the 1,3,5,7. And got 2, 8,48,384. Making 4,6,8,10. Interesting! Back to the "2 difference, but now the numbers are even instead of odd. Is that weird or am I doing something wrong? 
June 13th, 2018, 06:30 PM  #5 
Senior Member Joined: Jul 2012 From: DFW Area Posts: 621 Thanks: 86 Math Focus: Electrical Engineering Applications 
Hi Justin, What I was getting at (and I admit that I am not very clear most of the time) is, if we take 1,3,5,7,9 to calculate x^1, (instead of 1, 2, 3, etc.) in your first blue column we would get (with differences): $\displaystyle \begin{array}{cccccccccc} 1 & & 3 & & 5 & & 7 & & 9 \\ & 2 & & 2 & & 2 & & 2 & & =2^1 \cdot 1! \end{array}$ If we take 1,3,5,7,9 to calculate x^2, (instead of 1, 2, 3, etc.) in your second blue column we would get (with differences): $\displaystyle \begin{array}{cccccccccc} 1 & & 9 & & 25 & & 49 & & 81 \\ & 8 & & 16 & & 24 & & 32 \\ & & 8 & & 8 & & 8 & & & =2^2*2! \end{array}$ If we take 1,3,5,7,9 to calculate x^3, (instead of 1, 2, 3, etc.) in your third blue column we would get (with differences): $\displaystyle \begin{array}{cccccccccc} 1 & & 27 & & 125 & & 343 & & 729 \\ & 26 & & 98 & & 218 & & 386 \\ & & 72 & & 120 & & 168 & & & \\ & & & 48 & & 48 & & & & =2^3*3! \end{array}$ I'll let you work out using a difference of 3 for x^1, x^2, or x^3 if you are so inclined, but frankly, I probably think the result is more interesting than just about anybody else because I used it in some recent work. 

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