# Happy Birthday Ramanujam

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Today is 128th birthday of Srinivasa Ramanujam Iyengar.

[Please note that Britishers gave wrong English spelling for his name, they called him “Ramanujan”,  but it should be “Ramanujam” which means  “The younger brother of Lord Rama”]

John Littlewood said (as recalled by G. H. Hardy (1921), in “Obituary Notices: Srinivasa Ramanujan”, Proceedings of the London Mathematical Society 19, p. lvii) :

Every positive integer is one of Ramanujan’s personal friends.

So, let’s talk about numbers on his birthday.

128 is 7th power of 2 and leads to 4th Mersenne Prime ($2^7 - 1=127$ is a prime)

128 is the largest number which is not the sum of distinct squares. [https://oeis.org/A001422]

and here is my present for him:

RAMANUJAM’S MAGIC SQUARE

When I was in High School, I learned making such “Special Date Magic Squares” form P.K. Srinivasan’s (1924-2005) book regarding Ramanujam’s work. [see: https://nrich.maths.org/1380 ]

Magic sum of this square is 69. Thus all rows, columns and diagonals add up to the same total of 69 and today’s date has been placed in first row.

69 is a value of $n$ where $n^2$ and $n^3$ together contain each digit once. Since, $69^2=4761$ and $69^3 = 328509$

Ramanujam also studied “Partition Function”, this function gives the number of ways of writing the integer n as a sum of positive integers, where the order of addends is not considered significant. Observe that, in the magic square above we created 10 distinct 4-partitions of 69. (though 2376 distinct 4-partitions of 69 are possible!!)

Also, the total sum of numbers in our magic square is $69\cdot 4 = 276 = 1^5 + 2^5 + 3^5$

Once more, Happy Birthday Ramanujam!