### Today’s Puzzle:

A little more than a hundred years ago near Cambridge University G. H. Hardy took a taxi to visit his young friend and fellow mathematician, Srinivasa Ramanujan, in the hospital. Hardy couldn’t think of anything interesting about his taxi number, 1729, and remarked to Ramanujan that it appeared to be a rather dull number. But even the reason for his hospitalization could not prevent Ramanujan’s genius from shining through. He immediately recognized 1729’s unique and very interesting attribute: it is the SMALLEST number that can be written as the sum of two cubes in two different ways! Indeed,

12³ + 1³ = 1729, and

10³ + 9³ = 1729.

Today’s puzzle looks a little bit like a modern-day American taxi cab with the clues 17 and 29 at the top of the cab. The table below the puzzle contains all the Pythagorean triples with hypotenuses less than 100 sorted by legs and by hypotenuses. Use the table and logic to write the missing sides of the triangles in the puzzle. The right angle on each triangle is the only one that is marked. Obviously, none of the triangles are drawn to scale. No Pythagorean triple will appear more than once in the puzzle.

Here’s the same puzzle without all the added color:

Print the puzzles or type the solutions in this excel file: 10 Factors 1721-1729.

What taxi cab might Hardy have tried to catch next? He might have had to wait a long time for it, 4104.

16³ + 2³ = 4104, and

15³ + 9³ = 4104.

### Factors of 1729:

- 1729 is a composite number.
- Prime factorization: 1729 = 7 × 13 × 19.
- 1729 has no exponents greater than 1 in its prime factorization, so √1729 cannot be simplified.
- The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1729 has exactly 8 factors.
- The factors of 1729 are outlined with their factor pair partners in the graphic below.

### More About the Number 1729:

Did you notice these cool-looking factor pairs?

19 · 91 = 1729.

13 · 133 = 1729.

1729 is the hypotenuse of a Pythagorean triple:

665-1596-1729 which is **133** times (5-12-**13**).

1729 is the difference of two squares in four different ways:

865² – 864² = 1729,

127² – 120² = 1729,

73² – 60² = 1729, and

55² – 36² = 1729.

1729 is also a pseudoprime number! For example, even though it is a composite number, it passes the quick prime number test I first wrote about in 341 is the smallest composite number that gives a false positive for this Quick Prime Number Test:

It is known as a Carmichael number because back in 1909, Robert D. Carmichael, an American mathematician, showed that it passed even more prime number tests than this one.

𝗠𝘂𝗹𝘁𝗶𝗽𝗹𝗶𝗰𝗮𝘁𝗶𝗼𝗻 𝗯𝘆 𝟭𝟳𝟮𝟵#math1089 #math #maths #algebra #MULTIPLICATION #Hardy #Ramanujan pic.twitter.com/4wUNlujX8q

— Math1089 (@Math1089_9801) August 9, 2022

Can you find this famous taxi cab number which is also famously known as the Hardy-Ramanujan number? We think we have given enough hints now!!🤪🤓@raashibathija @DiveshNBathija #nationamathematicsday #ramanujan pic.twitter.com/SfUpn7eYFu

— UnMath School (@UnmathSchool) December 20, 2022

“I had ridden in taxi cab number 1729 remarking the number seemed to me a dull one like an unfavourable omen. “No Hardy,” he replied, “it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.”

– G.H. Hardy pic.twitter.com/U98c3S7aQy— UnMath School (@UnmathSchool) December 22, 2021

https://t.co/MbsbBoQEW6

*The Hardy-Ramanujan number 1729*“Iremember once going to see him when he was ill at Putney. I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen. pic.twitter.com/KuPt1fQPjl

— Vasavi Narayanan (@VasaviNarayanan) July 30, 2019

Hardy: I had ridden in taxi cab number 1729 and ..rather a dull one, and that I hoped… not a bad omen (factors 7*13*19) #Ramanujan: it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways.#NationalMathematicsDay pic.twitter.com/nObx998gvG

— Sudhanwa Pathak MBBS (@pathaksudh) December 22, 2018

Hardy -Ramanujan Taxi cab number. The number derives its name from the following story G.H. Hardy told about Ramanujan. “Once, in the taxi from London, Hardy noticed its number, 1729. He must have thought about it a little because he entered the room where Ramanujan lay in bed pic.twitter.com/XT8udVDxcX

— 𝐒𝐫𝐢𝐧𝐢𝐯𝐚𝐬𝐚 𝐑𝐚𝐠𝐡𝐚𝐯𝐚 ζ(1/2 + i σₙ )=0 (@SrinivasR1729) April 26, 2020

Read the complete thread about the astonishing story of ” Balaji, Ramanujan and Taxi No. 1729″. 🧵 1/8 pic.twitter.com/NcCjgvOxNX

— Jaspreet Bindra The Tech Whisperer (@j_bindra) January 8, 2023