Fermat's Last Theorem and ABC-Conjecture in the School of the XXI Century

In 1637, Pierre de Fermat wrote at the margins of Diophantus’ Arithmetica that he had found a truly wonderful proof of the insolvability of the Diophantine equation aⁿ + bⁿ = cⁿ, where n 2, but the narrow margins of the books did not allow him to give the full proof. Is there a short and easy way to prove Fermat's Last Theorem? The following ABC conjecture states that for three co-prime numbers A, B, and C which satisfy A + B = C, the product of the prime factors of ABC is usually not much less than C. Both theorems are formulated very simply, but are extremely difficult to prove. Hundreds of pages have been spent by eminent mathematicians of Western world searching for proofs, and the search for proofs continues. The author found new methods of proof that are generally understandable, even to schoolchildren on the basis of a synthesis of several sciences, including physics. Number theory plays an interesting role in pedagogy.