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# Primes Primos

Image taken from MathWorld.

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.

Note by Guilherme Dela Corte
3 years ago

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Level 1's answer is $$\boxed{1}$$ and Level 2's answer is $$\boxed{2}$$. How trivial.

- 3 years ago