The other day, Jim excitedly told me, "Did you realise I will turn x years old in the year x^2?"
He wasn't the first to think of this. The 19th century mathematician August de Morgan used to used to boast that he was x years old in the year x^2. He died in 1871.
In what year was Jim born? When will his prediction be true? In what year was de Morgan born? What is x in each case?
(In reply to answer
by K Sengupta)
De Morgan's Age
De Morgan, born in the 19th century was x years old in the year x^2
So, 1800< x^2 - x< 1900
Or, (2x-1) (- (84.8587, 87.18371)
Or, x (- (42.92935, 44.091855)
So, x= 43, 44
But for x = 44, we obtain 44^2 = 1936, implying that De Morgan was a mathematician of the 20th century, a contradiction.
Consequently, de Morgan was born in 1806 and he was precisely 1849 - 1806 = 43 years old in the year 1849 (43^2)
The only perfect square in the 20th century is 44^2 = 1936.
This would make Jim's birth year as 1936 - 44 = 1892, implying that Jim was born in the 19th century, contradicting the fact that Jim is a present day person.
The next perfect square occurs at 2025 (45^2), giving Jim's birthyear as 45^2 - 45 = 1980.
Accordingly, Jim was born in the year 1980 and the prediction will come true in the year 2025.