[caption id="attachment_3363" align="alignleft" width="418"] Google Doodle depicting Fermat's Last Theorem[/caption] We can find infinitely many solutions that can solve the equations $$x+y=z$$ and $$x^2+y^2=z^2$$ in integers, but what about the equation $$x^3+y^3=z^3$$ or more generally $$x^n+y^n=z^n$$, where n is an integer greater than 2 and x,y,...

Summary: Following the Indian (Hindu) calendar, we Indians are celebrating the seasonal festivals on wrong dates. It is because in the Indian calendars, the seasons are out of phase with the real tropical phenomenon of the earth. This article analyses how and why we are doing that and what to do about it. In the Indian calendars, the Makar Sankranti which marks the transition of the Sun into Makar Rasi (Capricorn), generally falls around 14th or 15th January of the Gregorian calendar. Makar Sankranti also marks many of the Indian harvest festivals such as the Pongal of the Tamils, the Bhogali Bihu of the Assamese, the Maghi (Lohri) of the Punabis, Bhogi in Andhra Pradesh etc. Many communities start their new years on this date. Astronomically, Makar Sankranti is the winter solstice. It is the shortest day marking the beginning of the Uttarayan (the northern journey) of the Sun with gradual increase of the duration of the day. The Bhagavad Gita mentions great importance of the Uttarayan. This was the reason why Bhishma, when wounded in Mahabharata war, chose to await for the Makar Sankranti, before choosing to die. In the Jagannath temple at Puri the Uttarayana Yatra is celebrated on this Makar Sankranti day.

The standard form of a linear equation in $$n$$ unknowns $$x_1,x_2,\dots ,x_n$$ is $$a_1x_1+a_2x_2+\dots +a_nx_n=b,$$ where $$a_1,a_2,\dots ,a_n$$ and $$b$$ are constants. Here constants mean some real numbers (these constants may come from any number field). A collection of one or more linear equations of same variables is called...