Binary or Septenary

Computer Science Level 5

Split the integers from 0 to 127 into two groups A and B such that sum of first, second, third, fourth, fifth and sixth powers of numbers in group A is equal to those in B, respectively.

i.e. ∑ ∀ A i A i p = ∑ ∀ B i B i p ∀ p ∈ { 1 , 2 , 3 , 4 , 5 , 6 } \text{i.e.} \sum_{\forall A_i}A_i^{p}= \sum_{\forall B_i}B_i^{p} \quad\quad\forall p\in\{1,2,3,4,5,6\}

Find the difference between their seventh power sum. That is, find

∣ ∑ ∀ A i A i 7 − ∑ ∀ B i B i 7 ∣ . \large\left|\sum_{\forall A_i}A_i^{7}- \sum_{\forall B_i}B_i^{7}\right|.

PS I don't know whether the solution I have is unique or not. So, if you get another solution and it satisfies conditions post it in this thread


The answer is 10569646080.

Rushikesh Jogdand by Rushikesh Jogdand , 22, India

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1 solution

David Hammond
Nov 9, 2019 
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# The Prouhet–Tarry–Escott problem
# @see https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence#The_Prouhet%E2%80%93Tarry%E2%80%93Escott_problem

N = 128

thue_morse = [bin(n).count('1') % 2 for n in range(N)]

Sa = [i for i in range(N) if thue_morse[i] == 1]
Sb = [i for i in range(N) if thue_morse[i] == 0]

def sum_power(S, power):
  return sum([s**power for s in S])

for p in range(1, 8):
  print(p, abs(sum_power(Sa, p) - sum_power(Sb, p)))

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