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Thursday, September 22, 2016

Entry 6

Ramanujan was a master at manipulating radicals. Some of his unusual ones are
353255275=5125+5325+5925 515+545=51125+52125+58125+516125 43+2453245=45+1451 81+1(1+52)24=1+521+452 The pattern of (2) is certainly suggestive, though I haven't seen any generalization of this. And (4) is the 8th root of an expression involving a 24th power! Also, note that he only uses ratios of pk with the small primes p=2,3,5. Either such simple relations are possible only for these, or Ramanujan found the low-hanging fruits and there are similar ones with prime p>5.

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