# What do Augustus De Morgan, Chelsea Clinton, Samuel Adams, and Caligula have in common?

The biography of Augustus De Morgan in The MacTutor History of Mathematics Archive ends with the following interesting tidbit.

De Morgan was always interested in odd numerical facts and writing in 1864 he noted that he had the distinction of being ${x}$ years old in the year ${x^{2}}$ (He was 43 in 1849). Anyone born in 1980 can claim the same distinction.

This got me thinking: how rare is this? Are there other birth years with this property?

Suppose this is true for a person who was born in the year ${b}$ and is ${y}$ years old. Then we would have ${y^{2}=y+b}$, the current year. (I’m assuming ${b>0}$, which of course I need not do!) Applying some fancy math we conclude that the person was born in the year ${b=y^{2}-y}$.

Thus, for each age, there is a corresponding birth year that has this property.

For example, ${45^{2}-45=1980}$. So a person born in 1980 (Chelsea Clinton, for instance) will be 45 in the year 2025 and ${45^{2}=2025}$.

(I should add that Chelsea Clinton would be able to say that the square of her age is the year only after she celebrates her birthday in 2025. Moreover, a person born in 1979 would be 45 years old in 2025 until his or her birthday in that year. So really, the birth years with this property are ${y^{2}-y}$ and ${y^{2}-y-1.}$)

Here are a few other people who could have made De Morgan’s claim (listed by year of birth).

1892: J.R.R. Tolkein was 44 in 1936
1806: John Stuart Mill was 43 in 1849
1722: Samuel Adams was 42 in 1764
1640: Bernard Lamy, the mathematician, was 41 in 1681
1560: Annibale Carracci, Italian painter was 40 in 1600
1482: Maria of Aragon and Castile, queen of Portugal would have been 39 in the year 1521 (she died in 1517)
1122: Eleanor of Aquitaine was 34 in the year 1156
12: Caligula was 4 in the year 16.

This sequence of birth years: 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110… has an entry in Sloane’s on-line encyclopedia of integer sequences. It is sequence number A002378 and it has the name “Oblong (or promic, pronic, or heteromecic) numbers: n(n+1).” But this instance of the sequence is not listed in the comment section. I’ll have to see if there is a way to submit items to the website.