While asleep, I had an unusual experience. There was a red screen formed by flowing blood, as it were. I was observing it. Suddenly a hand began to write on the screen. I became all attention. That hand wrote a number of elliptic integrals. They stuck to my mind. As soon as I woke up, I committed them to writing.
S. Ramanujan
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1729 is the natural number following 1728 and preceding 1730. It is commonly known as Ramanujan’s number and the Ramanujan-Hardy number.
This is a story about one of India’s great mathematical geniuses, S. Ramanujan. Once another famous mathematician Prof. G.H. Hardy came to visit him in a taxi whose number was 1729. While talking to Ramanujan, Hardy described this number a dull number. Ramanujan quickly pointed out that 1729 was indeed interesting. He said it is the smallest number that can be expressed as a sum of two cubes in two different ways:
1729 = 1728 + 1 = 123 + 13
1729 = 1000 + 729 = 103 + 93
Ramanujan knew the following formula for the sum of two cubes expressed in two different ways giving 1729, namely
(x2 + 9xy – y2)3 + (12x2 – 4xy + 2y2)3 = (9x2 – 7xy – y2)3 + (10x2 + 2y2)3
for x = 1 and y = 1.
1729 has since been known as the Hardy-Ramanujan Number, even though this feature of 1729 was known more than 300 years before Ramanujan.
Single Digit Representations
We can write the number 1729 using each of the digits 0 to 9 individually. The mathematical operations involved here are addition, subtraction, multiplication, division, exponentiation, factorial etc.
1729 written using 0
1729 = 0! + {(0! + 0! + 0!) × (0! + 0! + 0! + 0!)} 0!+0!+0!
1729 written using 1
1729 = (11 + 1)1+1+1 + 1
1729 written using 2
1729 = (2/2 + 2) × (22 + 2)2 + (2/2)
1729 written using 3
1729 = (3 × 3 + 3)3 + (3/3)
1729 written using 4
1729 = 4 × (4 × 44 + 44) + (4/4)
1729 written using 5
1729 = 55 × (5 × 5 − 5) + (55 − 5)/5 + 5
1729 written using 6
1729 = 6 × 6 × (6 × 6 + 6 + 6) + (6/6)
1729 written using 7
1729 = 7 × 7 × (7 × 7 − 7 − 7) + 7 + 7
1729 written using 8
1729 = 8 × (8 × (8 + 8) + 88) + (8/8)
1729 written using 9
1729 = 9 × 9 × 9 + 999 + (9/9)
Representation in Increasing and Decreasing Order of Digits
The number 1729 can be written using the digits 1, 2, . . . , 9 in various ways. Below are few examples.
1729 written using the digits 1 to 5
1729 = 123 − 4 + 5
1729 = 54 × 32 + 1
1729 written using the digits 1 to 6
1729 = 123 + (−4 + 5)6
1729 = 6 × (5 + 4) × 32 + 1
1729 written using the digits 1 to 7
1729 = 123 × (4 × 5 − 6) + 7
1729 = (7 − 6) × (54 × 32 + 1)
1729 written using the digits 1 to 8
1729 = −1 + (2 + 34 + 5) × 6 × 7 + 8
1729 = 8 − 7 + 6 × (5 + 4) × 32 × 1
1729 written using the digits 1 to 9
1729 = 12 − 3 + 45 − 6 + 78 × 9
1729 = (98 − 7) × (6 × 5 − 4 × 3 + 2 − 1)
1729 Using the Numbers from 1 to 10 and Reverse
The number 1729 can be written using the digits 1 to 10 in ascending and descending order.
1729 = 1 + 23 + [−4 + 56 + {(7 + 8)/√9}!] × 10
1729 = 10 × (98 + 7 + 65 + √4) + 32 × 1
Palindromic Representation of 1729
Palindromic number reads the same backward or forward. For example, 18081 is a palindrome. The number 1729271 is also a palindrome made from the digits of 1729. The representation below is in terms of digits of this palindrome.
1729 = 1 + 72 × (9 × 2 + 7 − 1)
Representation of 1729 Ending in Zero
The number 1729 can be written in such way that it ends with 0 and starts with any of the digits 4, 5, 6, 7, 8 or 9.
1729 = (4 × 3)2+1 + 0!
1729 = 54 × 32 + 1 × 0!
1729 = 6! − 5 − 4 − 3! + 210
1729 = (7 + 6) × (−5 − 4! × 3 + 210)
1729 = 8 − 7 + (6 + 5) × 43 + 210
1729 = 9 + 8 × (7 × 6 × 5 − 4 − 3 + 2 + 10)
1729 Using the Powers
1729 can be represented with the help of various powers of the digits.
1729 = 11 + 27 + 45 + 54 − 72
1729 = 19 − 21 + 45 + 54 + 92
1729 = 13 + 26 + 32 + 45 + 54 + 61
1729 = 03 + 10 + 26 + 32 + 45 + 54 + 61
1729 = 15 + 28 + 39 + 44 + 51 − 67 + 72 + 86 − 93
1729 = 04 + 17 + 29 − 38 + 46 + 55 + 62 + 71 + 83 + 90
1729 as the Difference of Two Squares
The number 1729 can be expressed in the form 𝑎2 − 𝑏2. The following are few examples.
1729 = 552 − 362
1729 = 732 − 602
1729 = 1272 − 1202
1729 = 8652 − 8642
1729 as a Sum of Squares
1729 = 62 + 182 + 372
1729 = 82 + 122 + 392
1729 = 82 + 242 + 332
1729 = 102 + 272 + 302
1729 = 122 + 172 + 362
1729 = 182 + 262 + 272
1729 as a Sum of Cubes
1729 = 13 + 123
1729 = 93 + 103
1729 =13 + 63 + 83 + 103
1729 = 13 + 33 + 43 + 53 + 83 + 103
1729 as a Product of Sum or Difference of Cubes
1729 = (63 − 53) × (33 − 23)
1729 = (43 + 33) × (33 − 23)
An Interesting Pattern involving 1729
Below is a mathematical pattern involving 1729. It comes in a very interesting way. In the first line, there is no 0 between two 1729, but we can see increasing number of 0’s between two 1729 in the succeeding lines (in fact, the number of 0’s is increased by 1). Also, the number 1729 is on both sides of the expression.
1729 × 10001 + 10001 × 9271 = 17291729 + 92719271
1729 × 100001 + 100001 × 9271 = 172901729 + 9271 09271
1729 × 1000001 + 1000001 × 9271 = 1729001729 + 9271 009271
1729 × 10000001 + 10000001 × 9271 = 17290001729 + 9271 0009271
Loeschian Quadratic Form
1729 is the lowest number which can be represented by a Loeschian quadratic form x2 + xy + y2 in four different ways with x and y positive integers. The integer pairs (x, y) are (25, 23), (32, 15), (37, 8) and (40, 3).
Single Letter Representation
The prime factors of the number 1729 are 7, 13 and 19. So, we can write the number as
1729 = 7 × 13 × 19 = 91 × 19 = 7 × 247 = 13 × 133.
Recall that, a three-digit number abc and a two-digit number de can be written as
abc = 102a + 10b + c and de = 10d + e.
Using this, the number 1729 is written in terms of each digit separately. In the following representation of 1729, b can take any value from 1 to 9.
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Reference:
Inder J. Taneja – Hardy-Ramanujan Number -1729