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Let \( \Sigma (m) \) be the sum of the first \( m \) prime numbers.

Level 1 - Evaluate \( \log_{10} \Sigma (3) \).

Level 2 - Evaluate \( \log_{10} \Sigma (3^2) \).

Level 3 - Evaluate the next pair of integers \( (m,n) \) such that the equation \( \Sigma (m) = 10^n \) holds.

Level 4 - Find a closed-form formula or a quick recurrence for \( \Sigma (m) \) that works perfectly for \( m > 10. \)

Level 5 - Are there infinitely many pairs of integers \( (m,n) \) such that the equation \( \Sigma (m) = 10^n \) holds? Prove or disprove this conjecture.

Computer Science answers are strongly recommended after **Level 3**. Manual solutions, however, will be accepted and rewarded with a million Zimbabwean dollars.

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TopNewestLevel 1's answer is \( \boxed{1} \) and Level 2's answer is \( \boxed{2} \). How trivial. – Guilherme Dela Corte · 2 years, 4 months ago

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