Le Monde puzzle [#1119]

December 7, 2019
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A digit puzzle as Le weekly Monde current mathematical puzzle that sounds close to some earlier versions:

Perfect squares are pairs (a²,b²) with the same number of digits such that a²b² is itself a square. What is the pair providing a²b² less than 10⁶? Is there a solution with both integers enjoying ten digits?

The run of a brute force R code like

```cek<-function(a,b){
u<-trunc
if ((n<-u(log(a^2,ba=10)))==u(log(b^2,ba=10))&
(u(sqrt(a^2*10^(n+1)+b^2))^2==(a^2*10^(n+1)+b^2))) print(c(a,b))}```

provides solutions to the first question.

```[1] 2 3
[1] 4 9
[1] 12 20
[1] 15 25
[1] 18 30
[1] 49 99
[1] 126 155
[1] 154 300
[1] 159 281
[1] 177 277
[1] 228 100
[1] 252 310
[1] 285 125
```

with the (demonstrable) conclusion that the only pairs with an even number of digits are of the form (49…9²,9…9²), as for instance (49999²,99999²) with ten digits each.

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