Tag Archives: Penguin Dictionary of Curious and Interesting Mathematics

Graham’s Sumber

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77

Every number greater than 77 is the sum of integers, the sum of whose reciprocals is 1.

For example, 78 = 2 + 6 + 8 + 10 + 12 + 40 and 1/2 + 1/6 + 1/8 + 1/10 + 1/12 + 1/40 = 1.

R.L. Graham, “A Theorem on Partitions”, Journal of the Australian Mathematical Society, 1963; quoted in Le Lionnais, 1983.

• From David Wells’ Penguin Dictionary of Curious and Interesting Mathematics (1986)

Post-Performative Post-Scriptum…

The title of this post is a pun on the gargantuan Graham’s number, described by the same American mathematician and famous among math-fans for its mindboggling size. “Le Lionnais, 1983” must refer to a book called Les Nombres remarquables by the French mathematician François Le Lionnais (1901-84).

Square Pairs

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Girard knew and Fermat a few years later proved the beautiful theorem that every prime of the form 4n + 1; that is, the primes 5, 13, 17, 29, 37, 41, 53… is the sum of two squares in exactly one way. Primes of the form 4n + 3, such as 3, 7, 11, 19, 23, 31, 43, 47… are never the sum of two squares. — David Wells, The Penguin Dictionary of Curious and Interesting Numbers (1986), entry for “13”.

Elsewhere other-accessible…

• Fermat’s theorem on sums of two squares
• Pythagorean primes

Triangular Squares

by in √2, Integer Sequences, Irrational numbers, Mathematics, Square root of 2, Square Roots, Squares, Triangular Numbers and tagged , , , , , , , , , , , , , | Leave a comment

The numbers that are both square and triangular are beautifully related to the best approximations to √2:

Number

Square Root

Factors of root
1 1 1
36 6 2 * 3
1225 35 5 * 7
41616 204 12 * 17

and so on.

In each case the factors of the root are the numerator and denominator of the next approximation to √2. — David Wells, The Penguin Dictionary of Curious and Interesting Mathematics (1986), entry for “36”.

Elsewhere other-accessible

• A001110 — Square triangular numbers: numbers that are both triangular and square