-How many formulas are there such as 1 + 2 ^ 7 = 127?

Some may have a special preference for formula 5 ^ 2 = 25-the same numbers are used on both sides of the equation, which is amazing. There are many similar formulas, such

5 ^ (6-2) = 625
(4/2) ^ 10 = 1024
(86 + 2*7) ^ 5-91)/3 ^ 4 = 123456789

We naturally raised the question: how many such formulas are there? The answer is: infinite. With the formula 5 ^ 2 = 25 mentioned at the beginning of this article, we can easily construct an infinite number of formulas that also satisfy this magical nature:

50 ^ 2 + 0 = 2500
500 ^ 2 + 0 + 0 = 250000
5000 ^ 2 + 0 + 0 + 0 = 25000000
......

Now, let's look at another more subtle formula: the order of numbers on both sides of the equation is the same!

-1 + 2 ^ 7 = 127
(3 + 4) ^ 3 = 343
16 ^ 3 * (8-4) = 16384

Is there an infinite number of such formulas?

 
The answer is still yes, and interestingly, its structure can still be extended by the classic formula 5 ^ 2 = 25. After slightly modifying the 50 ^ 2 + 0 = 2500 mentioned above, we can obtain an equation with the same numerical order on both sides:

2 + 50 ^ 2 = 2502

It can continue to derive an infinite number of statements that meet the requirements:

2 + (500 + 0) ^ 2 = 250002
2 + (5000 + 0 + 0) ^ 2 = 25000002
2 + (50000 + 0 + 0 + 0) ^ 2 = 2500000002
......

It can be seen that even if the order of numbers on both sides of the equation is the same, there are still an infinite number of matching formulas.

 
However, the above structures are only valid in decimal. In other hexadecimal systems, is this formula still infinite? This interesting topic was discussed in the uyhip puzzle last month. In fact, we only needOneThe clever construction can be explained that in the hexadecimal system used, such formulas are infinitely many. Calculation Formula

(M + 9/9) * (9 + 9/9) ^ (9 + 9/9)-9/9
= (M + 1) * 10 ^ 10-1
= M * 10 ^ 10 + 9999999999

Obviously, for any positive integer m, the numbers (including the order) used on the leftmost and rightmost sides of the equation are identical. We can easily modify this formula to apply it to any input system. For example, to get a formula under the eight-digit formula, you only need to replace all 9 in the formula with 7, and then change the index to 77 + 7/7 + 7/7 + 7/7 + .... Note that each 7/7 is added, so that two more 7 values in the formula are displayed, but only one 7 is displayed in the calculation result. Therefore, as long as the index is set to a number greater than 7 in the formula (for example, 77), 7/7 will be added after it, at one time, the calculation result is exactly the same as the number 7 in the formula.