# Square Root of 2 is Irrational, Proof 29

#### Cesare Palmisani,

22 December, 2017

The following is based on M. Jacobson and H. Williams' book.

We define two sequences of positive integers $(d_n)$ and $(s_n ).$ "The law of formation of these numbers is explained by Theon of Smyrna, and is as follows," see the reference:

(1)

$d_{n+1}=d_n+2s_n,\;d_1=1$

(2)

$s_{n+1}=s_n+d_n,\;s_1=1.$

The sequences (1) and (2) are called *diagonal numbers* and *side numbers*, respectively. Obviously the two sequences are monotone strictly increasing. In addition the sequences are generalized Fibonacci.

By induction,

(3)

$d_n^2-2s_n^2=(-1)^n,$

Let's suppose that $\sqrt{2}$ is a rational number. Then we can write $\displaystyle \sqrt{2}=\frac{a}{b},$ where $a,b$ are positive integers. From (3),

(4)

$(bd_n-as_n)(bd_n+as_n)=b^2\cdot (-1)^n.$

As a matter of fact,

$\displaystyle \begin{align} (bd_n-as_n)(bd_n+as_n)&=(bd_n)^2-(as_n)^2\\ &=b^2d_n^2-a^2s_n^2=b^2\left(d_n^2-\frac{a^2}{b^2}s^2_n\right)\\ &=b^2(d_n^2-2s_n^2=b^2\cdot (-1)^n. \end{align}$

If $bd_n-as_n=0,$ then, successively, $bd_n=as_n,$ $\displaystyle d_n=\frac{a}{b}s_n,$ $\displaystyle d_n^2=\frac{a^2}{b^2}s_n^2,$ $d_n^2-2s_n^2=2s_n^2-2s_n^2=0,$ in contradiction with (3). Therefore, $bd_n-as_n\ne 0,$ and we can divide in (4):

$\displaystyle bd_n+as_n=\frac{b^2\cdot (-1)^n}{bd_n-as_n}.$

Since $bd_n+as_n\gt 0,$ $bd_n+as_n=|bd_n+as_n|$ and

$\displaystyle bd_n+as_n=\left|\frac{b^2\cdot (-1)^n}{bd_n-as_n}\right|=\frac{b^2}{|bd_n-as_n|}.$

In addition, from $bd_n-as_n\ne 0,$ it follows that $|bd_n-as_n|\ge 1,$ $\displaystyle \frac{1}{|bd_n-as_n|}\le 1,$ $\displaystyle \frac{b^2}{|bd_n-as_n|}\le b^2,$ or, $bd_n+as_n\le b^2,$ implying that the sequences $(d_n)$ and $(s_n)$ are bounded. A contradiction.

### References

- Heath T., A History of Greek Mathematics, Oxford University Press, Vol I, 92
- Jacobson M., Williams H., Solving the Pell Equation, CMS books in Mathematics, Springer, 24- 25 (2009)
- Palmisani C., Side and diagonal numbers: additional results, to be appear in Periodico di Matematiche - Mathesis; No. 1 (2018)

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