Sift of Sundaram

The screen of Sundaram makes it possible to list the odd natural entireties not first thanks to arithmetic continuations placed in columns. It is based on the fact that by determining the whole of the composed odd numbers, one can deduce the unit from it from the prime numbers. The column number N has for first term (2n + 1) ² and as a reason R = 4n + 2 . Consequently, a number odd > 1, absent from this table, will be first. Indeed, let us consider two unspecified odd numbers:

$I\_n\; =\; 2\; \backslash \; cdot\; N\; +\; 1$

$I\_p=2\; \backslash \; cdot\; p\; +\; 1$

Then one can write that: $I\_p\; =\; 2\; \backslash \; cdot\; p\; +\; 1\; =\; 2\; \backslash \; cdot\; N\; +\; 1\; +\; 2\; \backslash \; cdot\; k$

Then the product is worth: $I\_n\; \backslash \; cdot\; I\_p\; =\; (2n+1)\; \backslash \; cdot\; (2p+1)\; =\; (2n+1)\; ^2\; +\; K\; \backslash \; cdot\; (4n+2)$

Thus, while varying N and K one obtains the whole of the products of two odd numbers which one reproduces in this table.

Sundaram was an Indian mathematician . The screen which it published in 1934 was a little different from the model above. It contained the values N such as 2n + 1 is not first. The table of this page offers directly the values 2n + 1 .

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