On August the 5 th I have posted on Tao's blog a link to my proof of the three consecutive non increasing inequalities between the differences of consecutive primes. This, I thought, was a question raised by Erdos and Turan in their 1948 paper "On some new questions on the distribution of prime numbers". On August 11 th Fan answer my post with the following information : Erdos and Turan were asking for decreasing inequalities (not non increasing ones). This information was already at the end of my work but Fan pointed out that with non decreasing inequality my results can be obtained from the prime number theorem. In the end the value of my work is the "elementary" proof of these non increasing inequalities without use of the prime number theorem. Thanks again to Fan.