Consider the Diophantine equation x³ + y³ = p z³, where p is a prime and we seek non-trivial integer solutions (x,y,z) ∈ ℤ³ \setminus (0,0,0). By a classical descent argument using the fact that ℚ(√-3) has class number 1, one can show that the equation x³ + y³ = z³ has only trivial solutions (Fermat's Last Theorem for n=3). For which primes p does the generalized equation admit non-trivial solutions? I suspect that p ≡ 1 (mod 3) and p = 3 are candidates. Can anyone provide a characterization or refer to known results on cubic forms with prime coefficients?

I am studying AP Calculus BC and I keep going in circles when trying to evaluate this integral: ∫ e^x sin(x) dx I tried using integration by parts with u = e^x and dv = sin(x)\,dx, which gave me: ∫ e^x sin(x)\,dx = -e^xcos(x) + ∫ e^xcos(x)\,dx Then I applied integration by parts again on ∫ e^xcos(x)\,dx and ended up back where I started. Could someone explain the standard technique for solving this type of cyclic integral?

I am studying abstract algebra and I need to understand the structure of subgroups of cyclic groups. The theorem: Every subgroup of a cyclic group is cyclic. Moreover, for G = ⟨ g ⟩ of order n, for each divisor d of n, there is exactly one subgroup of order d, generated by g^(n/d). My questions: 1. How do I find all subgroups of ℤ_(12) (additive group modulo 12)? 2. How do I draw the subgroup lattice diagram? 3. What changes if G is infinite cyclic (ℤ)? 4. How does this theorem help in understanding Euler's totient function φ(d) — the number of generators of a cyclic group of order d? 5. How many elements of order 6 are there in ℤ₆ × ℤ₆? I want to see the complete subgroup structure.

I am studying number theory and I need to understand the Chinese Remainder Theorem (CRT). Theorem: If m₁, m₂, \ldots, m_k are pairwise coprime, then the system: \begin{align*} x &\equiv a_1 \pmod{m_1} \\ x &\equiv a_2 \pmod{m_2} \\ &\vdots \\ x &\equiv a_k \pmod{m_k} \end{align*} has a unique solution modulo M = m₁ m₂ ⋯ m_k. My questions: 1. Why does CRT work intuitively? 2. How do I compute the solution for: x ≡ 2 (mod 3), x ≡ 3 (mod 5), x ≡ 2 (mod 7) 3. How is CRT used in RSA decryption to speed up computations? 4. What happens if the moduli are NOT coprime? How do I handle it? 5. How does CRT generalise to rings (the CRT for rings)? I want the constructive proof with step-by-step computation.

I'm studying linear algebra and I need a clear method for checking whether a set of vectors is linearly independent. Formally, vectors v₁, v₂, \ldots, vₙ are linearly independent if the only solution to: c₁v₁ + c₂v₂ + ⋯ + cₙvₙ = 0 is c₁ = c₂ = ⋯ = cₙ = 0. But in practice, what is the fastest way to check this? I've heard about putting vectors in a matrix and computing the determinant or row reducing. Can someone explain the connection between these methods?

I am studying number theory and I need to understand Diophantine equations — equations where only integer solutions are allowed. Consider: 3x + 5y = 17 1. How do I determine if integer solutions exist? 2. If they exist, how do I find all solutions? 3. How do I restrict to non-negative solutions? 4. What is the general method for solving ax + by = c where a, b, c ∈ ℤ? I know the condition involves gcd(a,b) c, but I need the full solution method with an example.