An example of a pattern I thought would hold is Waring's Problem. The theorem is that for any natural numbers $k$ and $n$ if $n>n_0$ there is a $l$ such that $n$ is expressible as the sum of $l$ $k$-th powers. The patter comes in when we attempt to compute these numbers. Obviously any number is the sum of $1$ first power. Lagrange's Theorem shows that any $n$ is the sum of $4$ squares. Also, any large enough $n$ is the sum of $9$ cubes. One may be tempted to think that we would want $16$ fourth powers, however, this is where the pattern diverges. We actually need only $19$, and so onwe need $37$ fifth powers, and $73$ sixth powers.
An example of a pattern I thought would hold is Waring's Problem. The theorem is that for any natural numbers $k$ and $n$ if $n>n_0$ there is a $l$ such that $n$ is expressible as the sum of $l$ $k$-th powers. The patter comes in when we attempt to compute these numbers. Obviously any number is the sum of $1$ first power. Lagrange's Theorem shows that any $n$ is the sum of $4$ squares. Also, any large enough $n$ is the sum of $9$ cubes. One may be tempted to think that we would want $16$ fourth powers, however, this is where the pattern diverges. We need only $19$, and so on.
An example of a pattern I thought would hold is Waring's Problem. The theorem is that for any natural numbers $k$ and $n$ if $n>n_0$ there is a $l$ such that $n$ is expressible as the sum of $l$ $k$-th powers. The patter comes in when we attempt to compute these numbers. Obviously any number is the sum of $1$ first power. Lagrange's Theorem shows that any $n$ is the sum of $4$ squares. Also, any large enough $n$ is the sum of $9$ cubes. One may be tempted to think that we would want $16$ fourth powers, however, this is where the pattern diverges. We actually need $19$, and we need $37$ fifth powers, and $73$ sixth powers.
An example of a pattern I thought would hold is Waring's Problem. The theorem is that for any natural numbers $k$ and $n$ if $n>n_0$ there is a $l$ such that $n$ is expressible as the sum of $l$ $k$-th powers. The patter comes in when we attempt to compute these numbers. Obviously any number is the sum of $1$ first power. Lagrange's Theorem shows that any $n$ is the sum of $4$ squares. Also, any large enough $n$ is the sum of $9$ cubes. One may be tempted to think that we would want $16$ fourth powers, however, this is where the pattern diverges. We need only $19$, and so on.