## Tuesday, 12 March 2013

### TUESDAY, 12 MARCH 2013

Today is the $71^{st}$ day of the year.

$71$ is prime.

$71$ is also an Emirp. An Emirp is a prime number whose digits, when reveresed, also form a prime number. In this case, of course, it is $17$. See A006567.

If one takes any four consecutive numbers, multiplies them together and adds one then the resulting number is a perfect square. Here are the first eleven of these calculations:
$(1 \times 2 \times 3 \times 4) + 1 = 25 = 5^2$
$(2 \times 3 \times 4 \times 5) + 1 = 121 = 11^2$
$(3 \times 4 \times 5 \times 6) + 1 = 361 = 19^2$
$(4 \times 5 \times 6 \times 7) + 1 = 841 = 29^2$
$(5 \times 6 \times 7 \times 8) + 1 = 1681 = 41^2$
$(6 \times 7 \times 8 \times 9) + 1 = 3025 = 55^2$
$(7 \times 8 \times 9 \times 10) + 1 = 5041 = 71^2$
$(8 \times 9 \times 10 \times 11) + 1 = 7921 = 89^2$
$(9 \times 10 \times 11 \times 12) + 1 = 11881 = 109^2$
$(10 \times 11 \times 12 \times 13) + 1 = 17161 = 131^2$
$(11 \times 12 \times 13 \times 14) + 1 = 24025 = 155^2$
$(12 \times 13 \times 14 \times 15) + 1 = 32761 = 181^2$
The sequence of roots of these calculations is $5, 11, 19, 29, 41, 55, 71, 89, ...$
Not suprisingly this sequence is a sequence at the On-Line Encyclopedia of Integer Sequences, it is A028387.
The sequence has a formula of $n + (n + 1)^2$. As can be observered, $71$ is the seventh member of the sequence and $71 = 7 + 8^2$