Carmichael Number

Primes are those which pass AKS primality test,

For pseudo-primes Carmichael raised the zest.

Integer factorization have some fundamental association,

Any prime factor of a Carmichael number have no repetition.

The search for pseudo-primes have various algorithms,

Because of RSA, Carmichael numbers obtain spatial attention.

It has been proved that all Carmichael numbers are odd,

The congruence relation makes this picture broad.

Alford et al. proved they are infinitely many ones,

The finding of smallest such number was a milestone.

At least three primes are always their factors,

The third Carmichael number is Hardy-Ramanujan number.

Their intimacy with prime is wondering Cryptographers,

Unlocking their secrets can open new prime chapters.

Circle

Circle attracts because of its symmetrical joint,

It is equidistant from a fixed point.

Euclid's Element was the finest creation of geometry,

A line passing through the centre is called the line of symmetry.

The width of the circle is called its diameter,

Half of the diameter is the radius, which is a useful parameter.

A circle divides its plane into three parts,

Drawing it from three points is an interesting start.

Try to fix the ratio of the area of a circle to a radius square,

The ratio of perimeter and diameter is also our ancient desire.

A tangent touches the circle at a unique point,

The radius and tangent perpendicularly meet at this unique joint.

Tangents drawn from an external point of the circle are equal,

They are adjacent sides of a cyclic quadrilateral.

To fit a circle in a square is an ancient quadrature problem,

It shows π irrationality and is useful in many theorems.

The chord and curve of the circle cover the segment area,

The radius and curve of a circle decide the sector's criteria.

Real Number

Division of integers is an early math problem,

We can solve it by Euclid division algorithm.

Euclid division algorithm computes highest common factor,

In this we apply Euclid division lemma till we get no remainder.

Factorization of integer remain a challenge for mathematicians,

A composite number has always a unique factorization.

Fundamental Theorem of Arithmetic is crucial for integers,

In Euclid's Elements, it was recorded earlier.

Prime factorization is useful to find LCM and HCF ,

HCF times LCM of two numbers is the product of numbers themselves.

Locating irrationals on number line show beautiful geometry,

For a prime p, It is interesting to prove square root p irrationality.

Irrational numbers characterize by its decimal expansion,

Decimal expansion of rational terminates or non-terminating repetition.

Cantor worked on the enumeration of real and rational numbers,

Combinatorics has so many reward-able conjectures.

Heron’s Formula

Area of a triangle is half of the base multiplied by height,
Calculating height in every triangular shape is difficult to fight.


Heron's was a famous mathematician from Egypt,
Many results of mensuration written in his manuscript.


He formulated sides of the triangle with its area,
Verify this from the Pythagoras criteria.


Archimedes might know this over two centuries earlier,
In some cases, Brahmagupta and Bretschneider raised the portiere.


Heron gave approximation rule of the square root of a number,
Babylonian kept this method with another nomenclature.