Hi i have discovered that

is the lcm of 1 to n raised to the power with base 2. For e.g. is since the lcm of 1,2,3,4,5 is 60 and so the value of is 1152921504606846976.]]>What did you think of my sum answer in reply to your question?

]]>The problem with mathematics is that it is inherently contradictory/paradoxical.

Consider this, before we can think about something like e.g a math problem....we need time, say 1 second. But 1 second is composed of an infinite amount of fractions of a second. So before we can pass the threshold of 1 second, we first have to pass all these smaller fractions of time....this is an impossibility due to the infinite amount of fractions of time between the zero moment and 1 second...so no thought is possible mathematically.

In fact life is not possible...since there was no time to create it.

]]>So what possible transformations are isometries of the plane? You might think that there were a whole bunch of them: rotations, reflections, and translations. (A reflection followed by a translation is sometimes called a glide reflection.) In fact the picture is simpler than that: it turns out that reflections and translations can be built up from rotations alone! A translation in a certain direction is simply a reflection in two axes perpendicular that direction, while a rotation about a point O is a reflection in two axes through O.

Hence any translation is a reflection in two axes perpendicular to the direction of translation whose distance apart is half the distance to be translated.Hence any rotation is a reflection in two axes through the centre of rotation whose angular separation is half the angle to be rotated through.

]]>It is always possible to slice a three-layered ham sandwich with a single cut of a knife in such a way that each layer of the sandwich is divided into two exactly equal halves by the cut.

The ham-sandwich theorem can be proved using the Borsukâ€“Ulam theorem.

]]>I learn that it is possible to make a tetradecahedron with just regular hexagons and squares.

It is also possible to make a truncated icosahedron with regular pentagons and hexagons.

Let , and suppose that is a continuously differentiable function on . Then:One of the many applications of this theorem is using it to show that:

by writing

where denotes the greatest integer part of x, and is the Euler-Mascheroni constant.]]>or, more precisely:

where M is the so-called Meissel-Merten's constant, with value approximately equal to 0.2614972128476427837554268386086958590516...

This result is known as Merten's second theorem.

]]>It has been a while now, how did everything go? Looked at Numberphile page, it looks pretty good.

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