Consecutive is unpredictable (Part 1).

Number Theory Level 5

Let A = 1 2 34 − 5 6 78 + 9 0 12 − 3 4 56 + 7 8 90 \large{A= 12^{34}-56^{78}+90^{12}-34^{56}+78^{90}}

How many decimal digits does A \large{A} have?

Details:

The answer might be < 0 <0 or > 0 >0

The powers are in a consecutive order of the first 10 10 digits: 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

What about part 2 2 ? Find it here


The answer is 171.

Adam Phúc Nguyễn by Adam Phúc Nguyễn , 20, Vietnam

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2 solutions

The number of decimal digits of A A is that of 7 8 90 78^{90} and we have:

n = ⌈ log ⁡ 10 7 8 90 ⌉ = ⌈ 90 log ⁡ 10 78 ⌉ = ⌈ 90 × 1.8920 ⌉ = ⌈ 170.2885 ⌉ = 171 n = \lceil \log_{10} 78^{90} \rceil = \lceil 90 \log_{10} 78 \rceil = \lceil 90 \times 1.8920 \rceil = \lceil 170.2885 \rceil = \boxed{171}

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bases = [12, 56, 90, 34, 78]
expos = [34, 78, 12, 56, 90]

total = 0

for i in range(5):
  sign = 1 if i%2 else -1

  total += sign*(bases[i]**expos[i])

print len(str(total)) - 1
# total is negative so subtract '-' in length

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