MathLabs

Problem 1

Let pp and qq be positive integers such that pq=1−12+13−14+⋯−11318+11319\dfrac pq=1-\dfrac12+\dfrac13-\dfrac14+\cdots-\dfrac1{1318}+\dfrac1{1319}. Prove that pp is divisible by 19791979.
Step 1 of 4: Rewrite the alternating sum
Previous stepNext step
S=∑j=113191j−2∑j=165912j=∑j=66013191jS=\sum_{j=1}^{1319}\dfrac1j-2\sum_{j=1}^{659}\dfrac1{2j}=\sum_{j=660}^{1319}\dfrac1j
Detailed analysis

Separate the negative even terms from the full harmonic sum. The even contribution is twice the harmonic sum through 659, leaving exactly the block from 660 through 1319.