Simon's age
Weekly Problem 3 - 2016
How many times has Simon's age changed from a square to a prime?
How many times has Simon's age changed from a square to a prime?
Challenge level
Problem
Simon has his $101^\text{st}$ birthday tomorrow. His age will change from a square ($100$) to a prime ($101$). How many times has this happened before in his life?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
There are two possible ways of doing this. You can check all the square numbers less than $100$ to see if next number is prime.
Therefore there are four previous occasions on which Simon's age has changed from a square to a prime.
The squares less than $100$ are $n^2$, for $0 \leq n < 10$.
If $n$ is odd, then $n^2$ is also odd, so $n^2 + 1$ is even. This is only a prime if it is $2$, so $n = 1$.
If $n$ is even, the results are tabulated below:
| Square Number | Next Number | Prime? |
| $0$ | $1$ | No |
| $1$ | $2$ | Yes |
| $4$ | $5$ | Yes |
| $9$ | $10$ | No |
| $16$ | $17$ | Yes |
| $25$ | $26$ | No |
| $36$ | $37$ | Yes |
| $49$ | $50$ | No |
| $64$ | $65$ | No |
| $81$ | $82$ | No |
The squares less than $100$ are $n^2$, for $0 \leq n < 10$.
If $n$ is odd, then $n^2$ is also odd, so $n^2 + 1$ is even. This is only a prime if it is $2$, so $n = 1$.
If $n$ is even, the results are tabulated below:
| Square Number | Next Number | Prime? |
|---|---|---|
| $0$ | $1$ | No |
| $4$ | $5$ | Yes |
| $16$ | $17$ | Yes |
| $36$ | $37$ | Yes |
| $64$ | $65$ | No |
This means there are four occasions where this has occurred previously: at ages $1$, $4$, $16$ and $36$.