JavaScript (Novice Programmer) mod(3^146, 293) among others returning the same incorrect values in Matlab and JS

First note that mod(3^146,293)=292. For some reason, inputting mod(3^146,293) in Matlab returns 275. Inputting Math.pow(3,146) % 293 in JS returns 275. This same error occurs (as far as I can tell) every time. This leads me to believe I am missing something obvious but cannot seem to tell what.

Any help is much appreciated.

Answer:1

As discussed in the answers to this related question, MATLAB uses double-precision floating point numbers by default, which have limits on their resolution (i.e. the floating point relative accuracy, eps). For example:

>> a = 3^146

a =

     4.567759074507741e+69

>> eps(a)

ans =

     7.662477704329444e+53

In this case, 3146 is on the order of 1069 and the relative accuracy is on the order of 1053. With only 16 digits of precision, a double can't store the exact integer representation of an arbitrary 70 digit integer.

An alternative in MATLAB is to use the Symbolic Toolbox to create symbolic numbers with a greater resolution. This gives you the answer you expect:

>> a = sym('3^146')

a =

4567759074507740406477787437675267212178680251724974985372646979033929

>> mod(a, 293)

ans =

292
Answer:2

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