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.

#NumberTheory #PrimeNumbers #SumofPrimes

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

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

Guilherme Dela Corte - 6 years, 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$