*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**.

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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 = 12

^{3}+ 1^{3}1729 = 1000 + 729 = 10

^{3}+ 9^{3}

Ramanujan knew the following formula for the sum of two cubes expressed in two different ways giving 1729, namely

(*x*^{2} + 9*xy* – *y*^{2})^{3} + (12*x*^{2} – 4*xy* + 2*y*^{2})^{3} = (9*x*^{2} – 7*xy* – *y*^{2})^{3} + (10*x*^{2} + 2*y*^{2})^{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 + 4^{4}) + (4/4)

**1729 written using 5**

1729 = 55 × (5 × 5 − 5) + (5^{5} − 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 = 12^{3} − 4 + 5

1729 = 54 × 32 + 1

**1729 written using the digits 1 to 6**

1729 = 12^{3} + (−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 + 4^{5} − 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 + 2^{3} + [−4 + 56 + {(7 + 8)/√9}!] × 10

1729 = 10 × (98 + 7 + 65 + √4) + 3^{2} × 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! + 2^{10}

1729 = (7 + 6) × (−5 − 4! × 3 + 210)

1729 = 8 − 7 + (6 + 5) × 4^{3} + 2^{10}

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 = 1^{1} + 2^{7} + 4^{5} + 5^{4} − 7^{2}

1729 = 1^{9} − 2^{1} + 4^{5} + 5^{4} + 9^{2}

1729 = 1^{3} + 2^{6} + 3^{2} + 4^{5} + 5^{4} + 6^{1}

1729 = 0^{3} + 1^{0} + 2^{6} + 3^{2} + 4^{5} + 5^{4} + 6^{1}

1729 = 1^{5} + 2^{8} + 3^{9} + 4^{4} + 5^{1} − 6^{7} + 7^{2} + 8^{6} − 9^{3}

1729 = 0^{4} + 1^{7} + 2^{9} − 3^{8} + 4^{6} + 5^{5} + 6^{2} + 7^{1} + 8^{3} + 9^{0}

**1729 as the Difference of Two Squares**

The number 1729 can be expressed in the form 𝑎^{2} − 𝑏^{2}. The following are few examples.

1729 = 55^{2} − 36^{2}

1729 = 73^{2} − 60^{2}

1729 = 127^{2} − 120^{2}

1729 = 865^{2} − 864^{2}

**1729 as a Sum of Squares**

1729 = 6^{2 }+ 18^{2 }+ 37^{2}

1729 = 8^{2 }+ 12^{2 }+ 39^{2}

1729 = 8^{2 }+ 24^{2 }+ 33^{2}

1729 = 10^{2} + 27^{2} + 30^{2}

1729 = 12^{2} + 17^{2} + 36^{2}

1729 = 18^{2} + 26^{2} + 27^{2}

**1729 as a Sum of Cubes**

1729 = 1^{3 }+ 12^{3}

1729 = 9^{3 }+ 10^{3}

1729 =1^{3} + 6^{3} + 8^{3} + 10^{3}

1729 = 1^{3} + 3^{3} + 4^{3} + 5^{3} + 8^{3} + 10^{3}

**1729 as a Product of Sum or Difference of Cubes**

1729 = (6^{3 }− 5^{3}) × (3^{3 }− 2^{3})

1729 = (4^{3 }+ 3^{3}) × (3^{3} − 2^{3})

**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 = **1729**0**1729** + 9271 09271

**1729** × 1000001 + 1000001 × 9271 = **1729**00**1729** + 9271 009271

**1729** × 10000001 + 10000001 × 9271 = **1729**000**1729** + 9271 0009271

**Loeschian Quadratic Form**

1729 is the lowest number which can be represented by a Loeschian quadratic form *x*^{2} + *xy* + *y*^{2} 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* = 10^{2}*a* + 10*b* + *c* and *de* = 10*d* + *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*