Question Number 100042 by bobhans last updated on 24/Jun/20

Answered by Rasheed.Sindhi last updated on 25/Jun/20

   16p=r^3 −1      16p=(r−1)(r^2 +r+1)      r−1=1,2,4,8,16,p   r=2,3,5,9,17,p+1 (possible values)              p=(((r−1)(r^2 +r+1))/(16))   ^• r=2⇒p =(7/(16))∉P   ^• r=3⇒p=((26)/(16))∉P   ^• r=5⇒p=((4×31)/(16))∉P   ^• r=9⇒p=((8×91)/(16))∉P   ^• r=17⇒p=((16×307)/(16))=307∈P   ^• r−1=p⇒r^2 +r+1=16         r^2 +r−15=0       ⇒r∉N⇒r−1∉P      p=307 (unique solution)

Commented bybobhans last updated on 25/Jun/20

thank you sir

Commented by1549442205 last updated on 27/Jun/20

Can make your solution become shorter by considering  divisor (r^2 +r+1).It is a odd number

Commented byRasheed.Sindhi last updated on 28/Jun/20

Good idea sir!  Better idea is:Don′t think of divisor:  r^2 +r+1∈O⇒r−1=16 ∧ r^2 +r+1=p  p=17^2 +17+1=307∈P  Thanks!