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 years old in the year (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 and is years old. Then we would have , the current year. (I’m assuming , which of course I need not do!) Applying some fancy math we conclude that the person was born in the year .

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

For example, . So a person born in 1980 (Chelsea Clinton, for instance) will be 45 in the year 2025 and .

(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 and )

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.

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When my kids were 5 and 10, I commented to my wife that she was the only one that year whose age wasn’t a multiple of 5. Her response was that it was OK, because the previous year I had been the only one in the family whose age wasn’t a perfect square.

Yes, we are physics geeks. And I wasn’t 40 when my first child was born.