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Twitter Numbers

Follow Matt Enlow (@CmonMattTHINK) on twitter and you see him post a #140digitnumber of the day. The 140 digit numbers follow some type of cool pattern. For example, one of his tweets were

Today’s #140digitnumber is the sum of the first 10^20 (one hundred quintillion) perfect sixth powers:

This one caught my eye because I will be teaching summations soon. So, I thought I would look at the sum of the perfect squares. In WolframAlpha can you type sum. I started using upper sums that were powers of 10 and then counting the number of digits in the sum. Do this enough times and you will notice that when the upper sum is 10^a the number of digits in the sum will be 3a. Why does this happen?

First, the number of digits in a number can be found by taking the base ten log of the number and then the ceiling. For example, 5^{500} would have ceil(500 \log_{10} (5)) number of digits. Knowing this we can find out how many digits there are in sum of squares. Using the sum of squares formula (1/6)n(n+1)(2n+1), take ceil(\log_{10} (1/6)n(n+1)(2n+1) and now we can quickly find the number of digits in a sum. Note that for students this would be a good time to recap log properties.

Second, I went to desmos.com and created a slider for different n values. I defined n to be n=b^a Here are a few n values that create 140 digit numbers. 2^{155}, 3^{98}, 6^{60}, 9^{49}, 13^{42} Some neat patterns begin to show. 46^{23}, 47^{23} are the first pair of consecutives with the same exponent. 87^{24}, 88^{24}, 89^{24} are the first trio. My favorite so far is 62^{26} I stopped searching for b=110 and started search for a=2, that is some number b squared that would make a 140 digit number. I found that (2*10^{23})^2 is one. Wow, this means that b value goes from 2 to at least 2*10^{23} One could create a program and find just how many n=b^a values there are. I only found 25 of them. Also, what would be the largest amount of consecutive b values – that is b^{a}, (b+1)^a, (b+2)^a, (b+3)^n...

Overall, I think that these Twitter numbers will offer some great exploration in the classroom. I plan on showing the sum of squares in class and then have students explore sum of cubes, fours, fives, and sixes. Or a student can create their own function to sum. I will update later on how it goes. Below you can click on the desmos graph to see the summation I explained earlier.

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