Question Number 67481 by Mohamed Amine Bouguezzoul last updated on 27/Aug/19
![p is a prime number such that (1+p)^p ≡2[7] find all k such that p≡k[42]](https://www.tinkutara.com/question/Q67481.png)
$${p}\:{is}\:{a}\:{prime}\:{number}\:{such}\:{that}\:\left(\mathrm{1}+{p}\right)^{{p}} \equiv\mathrm{2}\left[\mathrm{7}\right] \\ $$$${find}\:{all}\:{k}\:{such}\:{that}\:{p}\equiv{k}\left[\mathrm{42}\right] \\ $$
Commented by Rasheed.Sindhi last updated on 30/Aug/19
![Some values of k (1+2)^2 ≡2[7]⇒2≡2[42]⇒k=2 (1+17)^(17) ≡2[7]⇒17≡17[42]⇒k=17 (1+43)^(43) ≡2[7]⇒43≡1[42]⇒k=1 .....](https://www.tinkutara.com/question/Q67676.png)
$$\boldsymbol{{Some}}\:{values}\:{of}\:{k} \\ $$$$\left(\mathrm{1}+\mathrm{2}\right)^{\mathrm{2}} \equiv\mathrm{2}\left[\mathrm{7}\right]\Rightarrow\mathrm{2}\equiv\mathrm{2}\left[\mathrm{42}\right]\Rightarrow{k}=\mathrm{2} \\ $$$$\left(\mathrm{1}+\mathrm{17}\right)^{\mathrm{17}} \equiv\mathrm{2}\left[\mathrm{7}\right]\Rightarrow\mathrm{17}\equiv\mathrm{17}\left[\mathrm{42}\right]\Rightarrow{k}=\mathrm{17} \\ $$$$\left(\mathrm{1}+\mathrm{43}\right)^{\mathrm{43}} \equiv\mathrm{2}\left[\mathrm{7}\right]\Rightarrow\mathrm{43}\equiv\mathrm{1}\left[\mathrm{42}\right]\Rightarrow{k}=\mathrm{1} \\ $$$$….. \\ $$
Commented by Mohamed Amine Bouguezzoul last updated on 30/Aug/19
$${nice}\:{work}\:{but}\:{we}\:{want}\:{all}\:{values}\:{of}\:{k} \\ $$