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I seek for very sparse representations of positive integers. Let $$\mathbb N=\{0,1,2,\ldots\}\ \ \ \text{and}\ \ \ \mathbb Z^+=\{1,2,3,\ldots\}.$$ Recall that a polynomial $P(x,y)$ is integer-valued if $P(x,y)\in\mathbb Z$ for all $x,y\in\mathbb Z$.

Question 1. Is there an integer-valued quadratic polynomial $P(x,y)$ such that $\{P(x,y)+2^k:\ x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N\}=\mathbb Z^+$?

I note that the first positive integer not represented by $x^2+y(3y+1)/2+2^k\ (x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N)$ is $55832434$.

Question 2. Whether each positive integer $n$ can be written as $$\frac{x(3x+1)}2+\frac{y(7y+1)}2+2^k$$ with $x,y\in\mathbb Z$ and $k\in\mathbb N$?

This is positive for each $n=1,\ldots,2\times10^7$. See http://oeis.org/A343503. I ever proved that any $n\in\mathbb N$ can be written as $x^2+y(3y+1)/2+z(7z+1)/2$ with $x,y,z\in\mathbb Z$, and conjectured that $$\left\{x^4+\frac{y(3y+1)}2+\frac{z(7z+1)}2:\ x,y,z\in\mathbb Z\right\}=\mathbb N.$$

Your comments are welcome! In particular, Question 2 needs further check if you are good at computation.

I seek for very sparse representations of positive integers. Let $$\mathbb N=\{0,1,2,\ldots\}\ \ \ \text{and}\ \ \ \mathbb Z^+=\{1,2,3,\ldots\}.$$ Recall that a polynomial $P(x,y)$ is integer-valued if $P(x,y)\in\mathbb Z$ for all $x,y\in\mathbb Z$.

Question 1. Is there an integer-valued quadratic polynomial $P(x,y)$ such that $\{P(x,y)+2^k:\ x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N\}=\mathbb Z^+$?

I note that $55832434$ is the first positive integer not represented by $x^2+y(3y+1)/2+2^k$ $(x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N)$.

Question 2. Whether each positive integer $n$ can be written as $$\frac{x(3x+1)}2+\frac{y(7y+1)}2+2^k$$ with $x,y\in\mathbb Z$ and $k\in\mathbb N$?

This is positive for each $n=1,\ldots,2\times10^7$, see http://oeis.org/A343503. I ever proved that any $n\in\mathbb N$ can be written as $x^2+y(3y+1)/2+z(7z+1)/2$ with $x,y,z\in\mathbb Z$, and conjectured that $$\left\{x^4+\frac{y(3y+1)}2+\frac{z(7z+1)}2:\ x,y,z\in\mathbb Z\right\}=\mathbb N.$$

Your comments are welcome! In particular, Question 2 needs further check if you are good at computation.

I seek for very sparse representations of positive integers. Let $$\mathbb N=\{0,1,2,\ldots\}\ \ \ \text{and}\ \ \ \mathbb Z^+=\{1,2,3,\ldots\}.$$ Recall that a polynomial $P(x,y)$ is integer-valued if $P(x,y)\in\mathbb Z$ for all $x,y\in\mathbb Z$.

Question 1. Is there an integer-valued quadratic polynomial $P(x,y)$ such that $\{P(x,y)+2^k:\ x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N\}=\mathbb Z^+$?

I note that the first positive integer not represented by $x^2+y(3y+1)/2+2^k\ (x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N)$ is $55832434$.

Question 2. Whether each positive integer $n$ can be written as $$\frac{x(3x+1)}2+\frac{y(7y+1)}2+2^k$$ with $x,y\in\mathbb Z$ and $k\in\mathbb N$?

This is positive for each $n=1,\ldots,2\times10^7$. See http://oeis.org/A343503. I ever proved that any $n\in\mathbb N$ can be written as $x^2+y(3y+1)/2+z(7z+1)/2$ with $x,y,z\in\mathbb Z$, and conjectured that $$\left\{x^4+\frac{y(3y+1)}2+\frac{z(7z+1)}2:\ x,y,z\in\mathbb Z\right\}.$$

Your comments are welcome! In particular, Question 2 needs further check if you are good at computation.

I seek for very sparse representations of positive integers. Let $$\mathbb N=\{0,1,2,\ldots\}\ \ \ \text{and}\ \ \ \mathbb Z^+=\{1,2,3,\ldots\}.$$ Recall that a polynomial $P(x,y)$ is integer-valued if $P(x,y)\in\mathbb Z$ for all $x,y\in\mathbb Z$.

Question 1. Is there an integer-valued quadratic polynomial $P(x,y)$ such that $\{P(x,y)+2^k:\ x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N\}=\mathbb Z^+$?

I note that the first positive integer not represented by $x^2+y(3y+1)/2+2^k\ (x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N)$ is $55832434$.

Question 2. Whether each positive integer $n$ can be written as $$\frac{x(3x+1)}2+\frac{y(7y+1)}2+2^k$$ with $x,y\in\mathbb Z$ and $k\in\mathbb N$?

This is positive for each $n=1,\ldots,2\times10^7$. See http://oeis.org/A343503. I ever proved that any $n\in\mathbb N$ can be written as $x^2+y(3y+1)/2+z(7z+1)/2$ with $x,y,z\in\mathbb Z$, and conjectured that $$\left\{x^4+\frac{y(3y+1)}2+\frac{z(7z+1)}2:\ x,y,z\in\mathbb Z\right\}=\mathbb N.$$

Your comments are welcome! In particular, Question 2 needs further check if you are good at computation.

I seek for very sparse representations of positive integers. Let $$\mathbb N=\{0,1,2,\ldots\}\ \ \ \text{and}\ \ \ \mathbb Z^+=\{1,2,3,\ldots\}.$$ Recall that a polynomial $P(x,y)$ is integer-valued if $P(x,y)\in\mathbb Z$ for all $x,y\in\mathbb Z$.

Question 1. Is there an integer-valued quadratic polynomial $P(x,y)$ such that $\{P(x,y)+2^k:\ x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N\}=\mathbb Z^+$?

I note that the first positive integer not represented by $x^2+y(3y+1)/2+2^k\ (x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N)$ is $55832434$.

Question 2. Whether each positive integer $n$ can be written as $$\frac{x(3x+1)}2+\frac{y(7y+1)}2+2^k$$ with $x,y\in\mathbb Z$ and $k\in\mathbb N$?

This is positive for each $n=1,\ldots,2\times10^7$. See http://oeis.org/A343503.

Your comments are welcome! In particular, Question 2 needs further check if you are good at computation.

I seek for very sparse representations of positive integers. Let $$\mathbb N=\{0,1,2,\ldots\}\ \ \ \text{and}\ \ \ \mathbb Z^+=\{1,2,3,\ldots\}.$$ Recall that a polynomial $P(x,y)$ is integer-valued if $P(x,y)\in\mathbb Z$ for all $x,y\in\mathbb Z$.

Question 1. Is there an integer-valued quadratic polynomial $P(x,y)$ such that $\{P(x,y)+2^k:\ x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N\}=\mathbb Z^+$?

I note that the first positive integer not represented by $x^2+y(3y+1)/2+2^k\ (x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N)$ is $55832434$.

Question 2. Whether each positive integer $n$ can be written as $$\frac{x(3x+1)}2+\frac{y(7y+1)}2+2^k$$ with $x,y\in\mathbb Z$ and $k\in\mathbb N$?

This is positive for each $n=1,\ldots,2\times10^7$. See http://oeis.org/A343503. I ever proved that any $n\in\mathbb N$ can be written as $x^2+y(3y+1)/2+z(7z+1)/2$ with $x,y,z\in\mathbb Z$, and conjectured that $$\left\{x^4+\frac{y(3y+1)}2+\frac{z(7z+1)}2:\ x,y,z\in\mathbb Z\right\}.$$

Your comments are welcome! In particular, Question 2 needs further check if you are good at computation.