This paper presents 68 unsolved problems and conjectures in number theory, covering topics such as additive representation functions, the Erdős–Fuchs theorem, multiplicative problems, additive and multiplicative Sidon sets, hybrid problems, arithmetic functions, the greatest prime factor function, and mixed problems. The author, András Sárközy, reflects on his career and the importance of passing on knowledge to the next generation of mathematicians. He recalls his early work based on Paul Turán's problems and his collaboration with Paul Erdős, who inspired many of his theorems. He dedicates the paper to the memory of Erdős and Turán, hoping that young readers will find it as engaging as he did when reading their problem papers. The paper discusses additive representation functions, which count the number of ways a number can be expressed as the sum of elements from a set A. It presents results on the monotonicity of these functions under certain conditions. For example, it is shown that if a set A contains all positive integers from a certain point on, then the number of representations r(A, n) is monotone increasing. Conversely, if the complement of A grows faster than log N, then r₁(A, n) cannot be monotone increasing. Similarly, if A grows slower than N/log N, then r₂(A, n) cannot be monotone increasing. However, there exists an infinite set A where N - A(N) is large and r₂(A, n) is monotone increasing. The paper also raises two open problems: whether there exists an infinite sequence A with infinitely many missing positive integers such that r₁(A, n) is monotone from a certain point on, and whether there exists an infinite sequence A with lower (or upper) asymptotic density less than 1 such that r₂(A, n) is monotone increasing from a certain point on.This paper presents 68 unsolved problems and conjectures in number theory, covering topics such as additive representation functions, the Erdős–Fuchs theorem, multiplicative problems, additive and multiplicative Sidon sets, hybrid problems, arithmetic functions, the greatest prime factor function, and mixed problems. The author, András Sárközy, reflects on his career and the importance of passing on knowledge to the next generation of mathematicians. He recalls his early work based on Paul Turán's problems and his collaboration with Paul Erdős, who inspired many of his theorems. He dedicates the paper to the memory of Erdős and Turán, hoping that young readers will find it as engaging as he did when reading their problem papers. The paper discusses additive representation functions, which count the number of ways a number can be expressed as the sum of elements from a set A. It presents results on the monotonicity of these functions under certain conditions. For example, it is shown that if a set A contains all positive integers from a certain point on, then the number of representations r(A, n) is monotone increasing. Conversely, if the complement of A grows faster than log N, then r₁(A, n) cannot be monotone increasing. Similarly, if A grows slower than N/log N, then r₂(A, n) cannot be monotone increasing. However, there exists an infinite set A where N - A(N) is large and r₂(A, n) is monotone increasing. The paper also raises two open problems: whether there exists an infinite sequence A with infinitely many missing positive integers such that r₁(A, n) is monotone from a certain point on, and whether there exists an infinite sequence A with lower (or upper) asymptotic density less than 1 such that r₂(A, n) is monotone increasing from a certain point on.