# Mathematician:Pietro Antonio Cataldi

## Mathematician

Italian mathematician and philanthropist who taught mathematics and astronomy.

Worked on the development of perfect numbers and continued fractions.

Attempted in vain (as so many before and since) to prove Euclid's fifth postulate.

Supposed to have discovered the $6$th and $7$th Mersenne primes $M_{17}$ and $M_{19}$ in $1588$.

Italian

## History

• Born: April 15, 1548, Bologna, Italy
• Died: February 11, 1626, Bologna, Italy

## On Mersenne Primes

Cataldi is supposed to have discovered the 6th and 7th Mersenne primes $M_{17}$ and $M_{19}$ in $1588$ by a brute-force method. Recent researches[1], however, suggest that these may have already been discovered by $1460$. But as no evidence has been found from that date that they had been proven to be prime, it is possible that these were just lucky guesses.

Cataldi also claimed the primality of the Mersenne numbers $M_{23}, M_{29}, M_{31}$ and $M_{37}$.[2] Fermat proved him wrong about $M_{23}$, which has $47$ as a divisor, and $M_{37}$ which has $223$ as a factor. Euler showed in 1738 that $M_{29}$ has the factor $233$. However, by $1772$ Euler had shown that $M_{31}$ is indeed prime.

As Cataldi had not actually demonstrated the primality of $M_{31}$ (and because of his mistakes regarding $M_{23}, M_{29}$ and $M_{37}$), he is not credited with its discovery -- that one goes fair and square to Euler.

## Publications

• 1602 -- 1617: Practica aritmetica (in four parts)
• 1603: Trattato de nvmeri perfetti di Pietro Antonio Cataldo
• 1603: Operetta delle linee rette equidistanti et non equidistanti (an attempted proof of Euclid's Parallel Postulate)
• 1611: Transformatione geometrica
• 1612: Trattato della quadratura del cerchio dove si esamina un nuovo modo di quadrarlo per numeri. Et insieme si mostra come, Dato un rettilineo, si formi un curvilineo equale ad esso Dato. Et di più alcune transformationi di curvilinei misti fra loro (on squaring the circle)
• 1613: Trattato del modo brevissimo di trovar la radice quadra delli numeri (on continued fractions)
• 1618: Operetta di ordinanze quadre (Military applications of algebra)

## References

1. Ettore Picutti, in Historia Mathematica pp 123 - 136 (1989).
2. See Chris Caldwell's website The Prime Pages, in particular The Largest Known Prime by Year.