# Tagged Questions

Questions on finding integer/rational solutions of equations.

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### Math contest integer triplet problem

Can any one help me with this? Determine all integer triples (x,y,z) such that 1 ≤ x ≤ y ≤ z and x + y + z + xy + yz + xz = xyz − 1. I thought of Vietta's formula but don't let me lead you into a ...
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### On seventh powers $x_1^7+x_2^7+\dots+x_n^7 = 2$?

We have, $$(-6m^3 + 1)^3 + (6m^3 + 1)^3 + (-6m^2)^3 = 2$$ $$(-8m^5 + 1)^5 + (8m^5 + 1)^5 + (-8m^6 + 2m)^5 + (-8m^6 - 2m)^5 + 2(8m^6)^5=2$$ The first identity has been long known, while the second ...
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### Modified Pell equation: $x^2-D y^2 = m$, $m\neq1$.

How does one solve the Diophantine equation $$x^2-Dy^2=m,$$ where $m$ is some fixed arbitrary integer? I understand that given the fundamental solution to $r^2-D s^2=1$, and any solution to the ...
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### Is the assertion about the form $\alpha x+\beta xy+\gamma y$ true?

In my answer, I was led to conjecture the following: Statement: If $\gcd(\alpha,\beta,\gamma)=1,$ then every integer can be written as $\alpha x+\beta xy+\gamma y$ for integer $x$ and $y$. ...
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### A diophantine equation related to primes.

I have $2$ prime numbers $p_1$ and $p_2$. I have to find the solution of $\large{p_1t_1+p_2t_2=1}$ where $t_1$ and $t_2$ are integers. How do I do this?
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### Solve $(a^2-1)(b^2-1)=\frac{1}4 ,a,b\in \mathbb Q$

Does the equation $(a^2-1)(b^2-1)=\dfrac{1}4$ have solutions $a,b\in \mathbb Q$? I search $0<p<1000,0<q<1000$, where $a=\dfrac{p}q$, but no solutions exist. I wonder is this equation ...
Is there any solution to the following system of diophantine equations? $$\left\{\begin{array}{l} 2.a^2 = b^2+c^2+d^2 \\ a^2 = e^2+f^2+g^2 , & \mbox{with }((a,b,c,d,e,f,g)>2)\in N\mbox{ and ... 1answer 138 views ### Does a^6+b^6 = c^6+d^3 have a non-trivial solution? It is conjectured that,$$x_1^8+x_2^8+x_3^8 = y_1^8+y_2^8+y_3^8\tag{1}$$has no non-trivial solutions. However, if we relax it a bit then,$$x_1^8+x_2^8+x_3^4 = y_1^8+y_2^8+y_3^4\tag{2}$$can be ... 0answers 49 views ### Abelian SubGroup Variant: Consider the following problem: Find integers x_1, x_2, x_3,\dots, x_n Such that:$$P(x_1,x_2,\dots, x_n) = Q$$for some integer Q and polynomial P where for all permutations of any set of ... 2answers 689 views ### Diophantine equation involving prime numbers : p^3 - q^5 = (p+q)^2 Find all pairs of prime nummbers p,q such that p^3 - q^5 = (p+q)^2. It's obvious that p>q and q=2 doesn't work, then both p,q are odd. Assuming p = q + 2k we conclude, by the equation, ... 2answers 185 views ### Solve x^2+y^2=2 for x,y\in\mathbb Q. Solve x^2+y^2=2 for x,y\in\mathbb Q. I think the answer should be in terms of 1 integer variable \in\mathbb Z only. I rewrite the equation to (x+y)^2+(x-y)^2=2^2, then by the formula of ... 1answer 915 views ### The Diophantine equation x^2 + 2 = y^3 How to solve the Diophantine equation x^2 + 2 = y^3 with x,y>0 ? (x,y are integers.) 1answer 52 views ### Intersection of two series I'm looking for the solutions to$$\begin{eqnarray} z &=& x_0 + i \Delta_x \\ z &=& y_0 + j \Delta_y \\ i &\ge& 0\\ j &\ge& 0 \end{eqnarray}$$Well, I know that ... 1answer 383 views ### On an exponential diophantine equation I am trying to find all integer solutions of 5^x + 12 ^y = 13^z. The obvious (and pursued) solution is (2, 2, 2), and no others. I've tried to use an appropriate modular arithmetic, but to no ... 1answer 105 views ### On the Pell-like Ax^2-By^2 = 1 This is connected to the post, Mere coincidence? (prime factors). I was looking at NeuroFuzzy's dataset and noticed the line, {{{1, {4, 2}}, {1, 4, 2, 4, 2}, 23762}} It seems this could be ... 1answer 231 views ### Linear equation with prime coefficient. Suppose we have a linear equation with two variables say x and y and three integer coefficient a , b and c (constant), where a and b are prime all are greater than zero. ax+by=c how ... 3answers 246 views ### Prove that there are exactly 16 solutions to this problem. Show that are are only 16 integer solutions to the following equation:$$11x + 8y + 17 = xy$$What I tried: I took a modulo 2, and I got that y must be even and x must be odd. But beyond that, I ... 2answers 236 views ### Find all integer solutions to Diophantine equation x^3+y^3+z^3=w^3 Compute all integer solutions to the equation$$x^3+y^3+z^3=w^3$$2answers 43 views ### How to solve parameters given 3 different types of information? How Do I solve this eqns?$$x+y+z = Axyz = Bx^2+y^2+z^2 = C$$I have tried it in this way,,,$$yz = B/x = Py+z = A-x = Qy(Q-y) = P\implies y^2-Qy+p = 0$$I can't figure ... 1answer 114 views ### Prove \forall a,b,k \in \Bbb Z^+ such that a \equiv -1 \bmod 3 and b \equiv 1 \bmod 3, 2^{2k-1}a,2^{2k}b are non-trivial polygonal numbers Below is my original question, which has since been modified to a more general form. Prove that \forall p,q \in \Bbb P and k \in \Bbb Z^+ such that q \equiv -1 \bmod 3 and p \equiv 1 \bmod 3, ... 1answer 92 views ### 4-dim. generalization of ab+ac+bc=0 The equation ab+ac+bc=0 can be parameterized by (a,b,c)=\lambda(-pq, p(p+q), q(p+q)). Is there a (similar) parameterization for ab+ac+ad+bc+bd+cd=0? What about the 5-dimensional case? Edit: ... 0answers 56 views ### A diophantine definition of the Kleene star Let f(x \, | \, y_1, \dots, y_n) be a Diophantine polynomial that generates the Diophantine set F. By Matiyasevich, the set F^* (Kleene star of F) is also Diophantine. My question: how can ... 2answers 158 views ### Number theory problem, 3rd degree diophantine equation How many positive integers are there that can be written in the form$$\frac{m^3+n^3}{m^2+n^2+m+n+1}$$where m and n are positive integers. I invented this problem and was stuck with it for a ... 1answer 194 views ### If x,y,z \in \mathbb{Z} such that x^4+y^4+z^4 \equiv 0 \pmod{29}, prove that x^4+y^4+z^4 \equiv 0 \pmod{29^4} If x,y,z \in \mathbb{Z} such that x^4+y^4+z^4 \equiv 0 \pmod{29}, prove that x^4+y^4+z^4 \equiv 0 \pmod{29^4}. I have no idea where to start, but this is my abstract algebra homework, so I ... 3answers 153 views ### Unique Integer solution of a non-linear equation How to find the integer solution of the equation$$\frac{m^2 + 2mn + n^2 -3m -n+2}{2}=2$$I know that there is a unique solution 1answer 157 views ### Find positive integer that can't be expressed by \lfloor x/2 \rfloor + y + xy Consider the expression$$\lfloor x/2 \rfloor + y + xy$$where x and y are positive integers, \lfloor x/2 \rfloor means rounding down to integer, for example, \lfloor 3/2 \rfloor = 1. Some ... 3answers 106 views ### algebraic geometry and elliptic curves Does ax^2+by^2=cz^2 have positive integer solutions? I know that the solution exists when (a,b,c)=(1,1,1) or (1,1,n^2+1), but I failed to produce a general formula. Any help would be ... 1answer 72 views ### Diophantine equation and cyclicity of \mathbb{F}_p^* I am trying to prove that the diophantine equation$$1998^2x^2+1997x+1995-1998x^{1998}=1998y^4+1993y^3-1991y^{1998}-2001y$$has no solution in integers (given that 1997 is a prime). To do so, ... 1answer 150 views ### Given p, m, how many r, k exist such that \sum_{i=0}^k{m+i \choose p} = {m+r \choose p}? I know that {m+1 \choose p+1} = {m \choose p} + {m \choose p+1}, does this identity extend further out? My guess is that there exist certain k such that there exists r > k where the title ... 2answers 232 views ### Pythagorean Quadruples: Consider the set of integers x_1, x_2, x_3, x_4 Such that:$$x_1^2 + x_2^2 + x_3^2 = x_4^2$$How does one compute all the solutions to this system? I have the following method in place for ... 2answers 111 views ### Step in a solution of y^2 = x^3 - 2 I am reading Algebraic Number Theory notes here by Keith Conrad. In page 9, there is a solution of y^2=x^3-2 using unique factorization in \mathbb{Z}[\sqrt{-2}]. We start by writing ... 0answers 44 views ### Gap:\;\;L(90,28) := {\{n90 - m28 ∈ N, n, m ∈ N, n < 28\}} Which elements of the sets Gap:$$L(90,28) := {\{n90 - m28 ∈ N, n, m ∈ N, n < 28\}}$$What would be a quick way to resolve? 1answer 59 views ### two questions about Diophantine Equation I am reading an article Modular Arithmetic by Richard Taylor. I have 2 questions: For which n, x^2+y^2=nz^2 has nontravial solutions? What are the solutions? A beautiful theorem of Hermann ... 1answer 499 views ### Prove that every practical number is either a power of two or a power of two times a non-trivial polygonal number Prove that every practical number is either a power of two or a power of two times a non-trivial polygonal number, where a number q is practical if and only if every integer less than or equal to ... 1answer 83 views ### A diophantine equation I want to understand why the equation U^2-(m^2-4)V^2=-4 (when U,V,m are odd number and m > 3) is impossible. (This came from a post I was reading here) 0answers 54 views ### Is this a fruitless approach to solving diophantine equations? Let P(X, Y, Z) be a polynomial over Q. Let's be concerned with integer solutions. Namely that there are no solutions (X,Y,Z) such that \gcd(X,Y) = 1. So let X,Y be coprime and arbitrarily ... 1answer 189 views ### Number of Solutions to Diophantine Equation (a) Let c < 2\pi be a positive real number. Show that there are inﬁnitely many integers n such that the equation x^2 + y^2 + z^2 = n has at least c\sqrt n integer solutions. (b) Find ... 1answer 341 views ### Number of teeth in gears I'm building something with an engine that uses gears to reduce/increse movement. The motor has itself some gears, and it's a stepper motor (it gives discrete steps), now the number of steps per ... 1answer 416 views ### A natural number equation For what values of n the equation x^2 - (2n+1)xy + y^2 + x = 0 has no solution in natural numbers ? (for n=1 it has a trivial solution). 2answers 122 views ### Diophantine equation x^3=a^2+b^2+c^2 Does anyone know if a formula exists to obtain all solutions of the above Diophantine equation? All numbers integers. Addendum: After seeing the answer from Tito Piezas III, I reconsidered the above ... 4answers 138 views ### The values of N for which N(N-101) is a perfect square For how many values of N (integer), N(N-101) is a perfect square number? I started in this way. Let N(N-101)=a^2 or N^2-101N-a^2=0. Now if the discriminant of this equation is a ... 3answers 610 views ### Infinite solutions of Pell's equation x^{2} - dy^{2} = 1 Let d > 1 be a squarefree integer. Prove that the equation x^{2} - dy^{2} = 1 has infinitely many solutions in \mathbb{Z} \times \mathbb{Z}. What I have done: let  \ \mathbb{K} = ... 0answers 125 views ### Count number of positive integer solutions of x^2(8x-3)=y^2z? Given the Diophantine equation$$ x^2(8x-3)=y^2z, $$is there a way to efficiently count the number of solutions that satisfy x+y+z\leq n, where n is a fixed given integer? Also, for any fixed ... 2answers 109 views ### The complete solution to x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k, for k=1,2? It's quite easy to give the complete rational solution to,$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k,\;\; \text{for}\; k=1,2\tag{1}$$One can express it in the form, ... 4answers 134 views ### Find a,b\in\mathbb{Z}^{+} such that \large (\sqrt[3]{a}+\sqrt[3]{b}-1)^2=49+20\sqrt[3]{6} find positive intergers a,b such that \large (\sqrt[3]{a}+\sqrt[3]{b}-1)^2=49+20\sqrt[3]{6} Here i tried plugging x^3=a,y^3=b (x+y-1)^2=x^2+y^2+1+2(xy-x-y)=49+20\sqrt[3]{6}  the right ... 2answers 134 views ### Equation representing all numbers Joe Roberts writes, in Lure of the Integers, that Matijasevič showed that "every integer has a representation in the form a^2+b^2+c^2+c+1". The citation he gives is Ju. V. Matijasevič, A ... 0answers 123 views ### Solving a particular system of Diophantine equations in n variables (Frobenius equations) I have a particular system of linear Diophantine equations in n variables for which I need to find all nonnegative integer solutions. Specifically, they are Frobenius equations, meaning the ... 1answer 272 views ### integer solutions of a^3 + b^2 = 100000 Find all integer solutions of a^3 + b^2 = 100000 ? I'm looking for one solution and get idea from that to write an analytic solution, but I've not found yet. Is it a good idea or I should start it ... 1answer 96 views ### How many solutions are possible to the equation a^x-b^y=c? If a,b,c\in \mathbb Z are known and a>b>1,(a,b)=1, how many integer solutions are possible to the equation$$a^x-b^y=c~?\tag1 Can $(1)$ has more than $4$ integer solutions ?
I'm reading a lot about the Erdös-Straus Conjecture (ESC), a conjecture that states that for every natural number $p \geq 2$, there exists a set of natural numbers $a, b, c$ , such that the following ...