1729 – The Hardy Ramanujan Number

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

Welcome to the blog Math1089 – Mathematics for All.

Glad you came by. I wanted to let you know I appreciate your spending time here at the blog very much. I do appreciate your taking time out of your busy schedule to check out Math1089!

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 + 9xyy2)3 + (12x2 – 4xy + 2y2)3 = (9x2 – 7xyy2)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.

Your suggestions are eagerly and respectfully welcome! See you soon with a new mathematics blog that you and I call Math1089 – Mathematics for All!“.

Reference:

Inder J. Taneja Hardy-Ramanujan Number -1729

2 comments

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: