0
votes
0answers
36 views

Diophantine like philosophy for computing trigonometric functions with approximation around intervals

I noticed that diophantine expressions are great to approximate constants or simple functions, as far as I know, they are not so great when it comes to approximate and compute transcendental functions ...
1
vote
1answer
45 views

Gap between smooth integers tends to infinity (Stoermer-type result)?

Consider the following claim : (*) Let $P$ be a finite set of primes, let $S$ be the set of natural numbers all of whose divisors are in $P$, and let $s_n$ denote the $n$-th element of $S$. Then ...
0
votes
0answers
26 views

Exponential Diophantine Inequality

how would one go about solving inequality of the form $|a2^n-b2^k|>1 $ for $a,b \in R$ and $n,k \in Z$. Assume that $|a|>|b|$. Any help will be appreciated. Thank you
13
votes
5answers
3k views

Fermat's Last Theorem near misses?

I've recently seen a video of Numberphille channel on Youtube about Fermat's Last Theorem. It talks about how there is a given "solution" for the Fermat's Last Theorem for $n>2$ in the animated ...
2
votes
1answer
89 views

Approximate solution of a diophantine equation

Consider the Diophantine equation $P(x)=y^2$, where $P$ is a (nonconstant) polynomial with integer coefficients and $x$ and $y$ must be integers. For $\varepsilon \gt 0$, I say that an integer $x$ is ...
0
votes
1answer
38 views

Why does a Norm Form have rational coefficients?

[From Wolfgang Schmidt's "Diophantine Approximation and Diophantine Equations"] "Let $K$ be a number field with $[K:\mathbb{Q}] = d$ and $L(X)$ be a linear form, say $L(X) = L(x_1,\dots,x_n) = ...
6
votes
1answer
224 views

Cube roots don't sum up to integer

My question looks quite obvious, but I'm looking for a strict proof for this. (At least, I assume it's true what I claim.) Why can't the sum of two cube roots of positive non-perfect cubes be an ...
2
votes
3answers
166 views

Sum of roots is integer

My question looks quite obvious, but I'm looking for a strict proof for this: Why can't the sum of two square roots of non-perfect squares be an integer? For example: $\sqrt8+\sqrt{15}$ isn't an ...
0
votes
0answers
70 views

Solving system of equations with mixed variable types

I'm looking for solutions to the non-linear system of equations $$ n_1x + (n_1 - 1)y = a_1 \\ n_2x + (n_2 - 1)y = a_2 \\ n_3x + (n_3 - 1)y = a_3 \\ n_4x + (n_4 - 1)y = a_4 $$ where $x$ and $y$ are ...
2
votes
0answers
162 views

Diophantine equations/Diophantine Geometry

I am very knew to this site and I am eagerly waiting for solutions of: (1) Let $x$ be an algebraic number with degree $n > 1$. Then there exists only finitely many rational numbers $p/q$ (in ...
0
votes
1answer
252 views

Ordered triplet query

$$x^2 + y^2 + z^2 = 3xyz$$ How many ordered triples $(x,y,z)$ are there that satisfy the above equation. are the only solutions $x=y=z=0$ and $1$? Are there non trivial solutions? I saw this ...
6
votes
2answers
535 views

Applying the Thue-Siegel Theorem

Let $p(n)$ be the greatest prime divisor of $n$. Chowla proved here that $p(n^2+1) > C \ln \ln n $ for some $C$ and all $n > 1$. At the beginning of the paper, he mentions briefly that the ...
7
votes
1answer
158 views

Diophantine equation $x^3+x+y^3+y = z^3 + z$ for $x,y,z>0$

Consider the Diophantine equation $x^3+x+y^3+y = z^3 + z$ for positive integer $x,y,z$. I tried small values and got some near equalities : $(5,6,7)$ and $(12,16,18)$ are true up to value $2$. $( 5^3 ...
0
votes
0answers
107 views

integer solutions to $a^m+nx^2 = y^n$ with various conditions

I consider the following equation with conditions of obtaining solutions $$a^m+nx^2 = y^n$$ This equation has solution when $a$ is an even prime and $x, y, m$ are positive integers with $(nx, y) = ...
1
vote
2answers
130 views

Constructive proof need to know the solutions of the equations

Observe the following equations: $2x^2 + 1 = 3^n$ has two solutions $(1, 1) ~\text{and}~ (2, 2)$ $x^2 + 1 = 2 \cdot 5^n$ has two solutions $(3, 1) ~\text{and}~ (7, 2)$ $7x^2 + 11= 2 \cdot 3^n$ has ...
3
votes
2answers
189 views

Solutions of $x^2 + 119 = 15 \cdot 2^n$ without trial and error

I seen this equation at math.stack exchange The equation $x^2 + 119 = 15 \cdot 2^n$ has only six solutions. Those are (1,3) (11, 4), (19, 5), (29, 6), (61, 8) and other one is I don't know. This ...
1
vote
2answers
378 views

Solutions of some Diophantine equations

Respected Mathematicians, The diophantine equation $2^x$ + $5^y$ = $z^2$ has solutions $x = 3, y = 0, z = 3$ and $x = 2, y = 1, z = 3$. I got these solutions by trial and error method. To be honest, ...