Its true or false? $1^7 + 7^7 + 4^7 + 1^7 + 7^7 + 2^7 + 5^7 < 1741725$

Question

I try to group this form:

$2 + (2 \times 7^7) + (3 \times 2^7) + 5^7$

but there is nothing what can I do after that.

Answer 1

$$1^7+7^7+4^7+1^7+7^7+2^7+5^7=1741725$$

Thus it is not less than $1741725$

It is equal to $1741725.$

How did you come up with this number?

Answer 2

After your step, since $2^7 = 128$ then we have $$2 + 2\times 7^7 + 3\times 128 + 5^7 < 1741725$$ and by subtracting both sides by $3\times 128 + 2 = 3\times(130 - 2) + 2 = 390 - 6 +2 = 386$, we get

$$2\times 7^7 + 5^7 < 1,741,725 - 386 = 1,741,725 - 400 + 14 = 1,741,325 + 10 + 4 = 1,741,339.$$

Then, since $5^7 = 78,125$, simply subtract that value from both sides to get $$2\times 7^7 < 1,741,339 - 78,125 = 1,663,214.$$ Now, divide both sides by $2$ to get $$7^7 < 1,663,214\div 2 = 831,607.$$ Now dividing both sides by $7^7$, we get that $1<1.00979183843$ which is true. Therefore, $$2 + 2\times 7^7 + 3\times 2^7 + 5^7 < 1,741,725.$$ But looking at the answer above, we see that you grouped the inequality incorrectly. In fact, $$2 +2\times 7^7+ 3\times 2^7 + 5^7 = 1,725,597.$$ Your error is in thinking that $4^7 = 2^7 + 2^7$. No. The right hand side is equal to $2\times 2^7 = 2^8$. Therefore $4^7 + 2^7 \neq 2^7 + 2^7 + 2^7 = 3\times 2^7$.