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    Porsche is a German automobile manufacturer founded in 1931 by Ferdinand Porsche. The brand is famous for producing high-performance sports cars, luxury vehicles, and innovative electric models.     Its most iconic model, the Porsche 911, debuted in 1964 and remains a symbol of speed, design, and engineering perfection. Today, Porsche’s lineup includes […]

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We start from a problem: Problem: Find the number of elements in such that it is congruent to module for some integer . Solution. The problem is the same as finding the number of solutions to     Using Chinese Remainder Theorem, the problem reduces to the system of equations         Our […]

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There is a problem in Real Analysis by Halsey L. Royden, Patrick M. Fitzpatrick. Suppose that outer-measure is defined by covering sets by countable collections of closed, bounded intervals rather than coverings by open, bounded intervals. Show that the outer- measure remains unchanged. I give my answer here. Proof. Let be a set of real […]

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In complex analysis, we have the following well-known theorem: Theorem If , then the Fourier inversion holds, namely, given     We have     The proof is not easy for a beginner who just touched this theorem. As for myself, I spent a tons of time on managing the proof. So I would like […]

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Our goal is to prove the following theorem by Carlo Sanna. The proof is adapted and changed slightly from his paper. Theorem. For any prime number and any , we have     Lemma 1 If is prime, , and , then     and     Proof. . Each of the sum is by […]

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The -th Harmonic number, denoted is defined by the sum of the reciprocals of the first  natural numbers:     This number has quite interesting properties. On thing we could consider when is finite is the quotient of the sum. If we write it as     Then many problems raise. Here, we have a problem […]

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