Nonexistence of Solutions to Certain Families of Diophantine Equations

In this work, I examine specific families of Diophantine equations and prove that they have no solutions in positive integers. The proofs use a combination of classical elementary arguments and powerful tools such as Diophantine approximations, Lehmer numbers, the modular approach, and earlier results proved using linear forms in logarithms. In particular, I prove the following three theorems. Main Theorem I. Let a, b, c, k ∈ Z+ with k ≥ 7. Then the equation (a^2cX^k − 1)(b^2cY^k − 1) = (abcZ^k − 1)^2 has no solutions in integers X, Y , Z \u3e 1 with a^2X^k ̸= b^2Y^k. Main Theorem II. Let L, M, N ∈ Z+ with N \u3e 1. Then the equation NX^2 + 2^L3^M = Y^N has no solutions with X, Y ∈ Z+ and gcd(NX,Y) = 1. Main Theorem III. Let p be an odd rational prime and let N, α, β, γ ∈ Z with N \u3e 1, α ≥ 1, and β, γ ≥ 0. Then the equation X^{2N} +2^{2α}5^{2β}p^{2γ} =Z^5 has no solutions with X, Z ∈ Z+ and gcd(X, Z) = 1

Nonexistence of Solutions to Certain Families of Diophantine Equations

In this work, I examine specific families of Diophantine equations and prove that they have no solutions in positive integers. The proofs use a combination of classical elementary arguments and powerful tools such as Diophantine approximations, Lehmer numbers, the modular approach, and earlier results proved using linear forms in logarithms. In particular, I prove the following three theorems. Main Theorem I. Let a, b, c, k ∈ Z+ with k ≥ 7. Then the equation (a^2cX^k − 1)(b^2cY^k − 1) = (abcZ^k − 1)^2 has no solutions in integers X, Y , Z \u3e 1 with a^2X^k ̸= b^2Y^k. Main Theorem II. Let L, M, N ∈ Z+ with N \u3e 1. Then the equation NX^2 + 2^L3^M = Y^N has no solutions with X, Y ∈ Z+ and gcd(NX,Y) = 1. Main Theorem III. Let p be an odd rational prime and let N, α, β, γ ∈ Z with N \u3e 1, α ≥ 1, and β, γ ≥ 0. Then the equation X^{2N} +2^{2α}5^{2β}p^{2γ} =Z^5 has no solutions with X, Z ∈ Z+ and gcd(X, Z) = 1

The Effect of Hints and Model Answers in a Student-Controlled Problem-Solving Program for Secondary Physics Education

Many students experience difficulties in solving applied physics problems. Most programs that want students to improve problem-solving skills are concerned with the development of content knowledge. Physhint is an example of a student-controlled computer program that supports students in developing their strategic knowledge in combination with support at the level of content knowledge. The program allows students to ask for hints related to the episodes involved in solving a problem. The main question to be answered in this article is whether the program succeeds in improving strategic knowledge by allowing for more effective practice time for the student (practice effect) and/or by focusing on the systematic use of the available help (systematic hint-use effect). Analysis of qualitative data from an experimental study conducted previously show that both the expected effectiveness of practice and the systematic use of episode-related hints account for the enhanced problem-solving skills of students

On a simple quartic family of Thue equations over imaginary quadratic number fields

Let $t$ be any imaginary quadratic integer with $|t|\geq 100$. We prove that the inequality $$|F_t(X,Y)| = | X^4 - t X^3 Y - 6 X^2 Y^2 + t X Y^3 + Y^4 | \leq 1$$ has only trivial solutions $(x,y)$ in integers of the same imaginary quadratic number field as $t$. Moreover, we prove results on the inequalities $|F_t(X,Y)| \leq C|t|$ and $|F_t(X,Y)| \leq |t|^{2 -\varepsilon}$. These results follow from an approximation result that is based on the hypergeometric method. The proofs in this paper require a fair amount of computations, for which the code (in Sage) is provided.Comment: 27 page

12.2-GHz methanol maser MMB follow-up catalogue - I. Longitude range 330 to 10 degrees

We present a catalogue of 12.2-GHz methanol masers detected towards 6.7-GHz methanol masers observed in the unbiased Methanol Multibeam (MMB) survey in the longitude range 330\circ (through 360\circ) to 10\circ. This is the first portion of the catalogue which, when complete, will encompass all of the MMB detections. We report the detection of 184 12.2-GHz sources towards 400 6.7-GHz methanol maser targets, equating to a detection rate of 46 per cent. Of the 184 12.2-GHz detections, 117 are reported here for the first time. We draw attention to a number of 'special' sources, particularly those with emission at 12.2-GHz stronger than their 6.7-GHz counterpart and conclude that these unusual sources are not associated with a specific evolutionary stage.Comment: accepted to MNRAS 21 Dec 201