It has long been known that restricting the diet of many kinds of animals made them live longer. The cause was more recently narrowed down to two sulfur-containing amino acids, methionine and cysteine, such that restricting only those extended the lifespan of test animals.

Experiments restricting primates' diet have failed to replicate the effect seen in other animals. Humans already live much longer than other mammals of similar size.

Very interesting, thank you. Are there any leading theories on why this works with some mammals but not primates?

There are many hypotheses, but no way discovered yet to check them.