1729

When Srinivasa Ramanujan, the great Indian mathematician, was ill with tuberculosis in a London hospital, his colleague G. H. Hardy went to visit him. Hardy, trying to initiate conversation, said to Ramanujan, "I came here in taxi-cab number 1729. That number seems dull to me which I hope isn't a bad omen."

"Nonsense," replied Ramanujan. "The number isn't dull at all. It's quite interesting. It's the smallest number that can be expressed as the sum of two cubes in two different ways." (Ramanujan recognized that \(1729 = 1^{3} + 12^{3}\) as well as \(9^{3} + 10^{3}\).)

153

"Simon Peter went up, and drew the net to land full of great fishes, an hundred and fifty and three: and for all there were so many, yet was not the net broken." (John 21:11)

Four interesting things about 153:

1. \(153 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17\)
2. \(153 = 1! + 2! + 3! + 4! + 5!\)
3. \(153 = 1^{3} + 5^{3} + 3^{3}\)
4. 153 lies dormant in every third number. Take any multiple of three, sum the cubes of its digits, take the result, sum the cubes of its digits, take the results, etc. You eventually get 153. Take 12, for example.
   \(1^{3} + 2^{3} = 9\)
   \(9^{3} = 729\)
   \(7^{3} + 2^{3} + 9^{3} = 1080\)
   \(1^{3} + 0^{3} + 8^{3} + 0^{3} = 513\)
   \(5^{3} + 1^{3} + 3^{3} = 153\)

3435

This number is equal to the sum of its digits each raised to a power equal to the digit.

\(3435 = 3^3 + 4^4 + 3^3 + 5^5\)

Thanks to Burhanuddin Salman for telling me about this number, which is apparently called a "Munchausen number".

220

In Genesis 32:14, Jacob gives Esau 220 goats ("two hundred she goats and twenty he goats") as a gesture of friendship.

The Pythagoreans identified 220 as a "friendly" number. That is, 220 has a close friend: 284. Each are equal to the sum of the proper divisors of the other. Proper divisors are all the numbers that divide evenly into a number, including 1 but excluding the number itself. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. Add all those numbers and you get 284. Likewise, the proper divisors of 284 are 1, 2, 4, 71, and 142 and they sum to 220.

Curious calculations

1. Pick any 3-digit whole number. (185)
2. Repeat the digits. (185,185)
3. Divide by 7 (\(185,185 \div 7 = 26,455\))
4. Divide by 11 (\(26,455 \div 11 = 2405\))
5. Divide by 13 (\(2405 \div 13 = 185 \)). Notice you get your original number.

1. Choose any prime number greater than 3. (Try 19)
2. Square that number (\(19 \times 19 = 361\))
3. Add 14 (\(361 + 14 = 375\))
4. Divide by 12 (\(375 \div 12 = 31\) with remainder 3). Notice the remainder will always be 3. (If you add 17 instead of 14, your remainder will always be 6.)

1. Pick any 3 digit number. (682)
2. Write this number backwards and subtract the smaller number from the other. (\(682 - 286 = 396\))
3. Take this answer and again invert it. (693)
4. Add your previous "answer" to its inverse (\(396 + 693 = 1089\)). Notice doing this with any three digit number will either produce 1089 or 0. Even single digits, when written 00x, will work. For example, Try 8. (\(800 - 008 = 792\). Then \(297 + 792 = 1089\))

Other curiosities

\(111,111,111 \times 111,111,111 = 12,345,678,987,654,321\)

\(1,741,725 = 1^{7} + 7^{7} + 4^{7} + 1^{7} + 7^{7} + 2^{7} + 5^{7}\)