Richert’s Theorem, proved by Hans-Egon Richert in 1948, states that every integer greater than or equal to 7 can be expressed as a sum of distinct prime numbers. The article presents a short proof based on Bertrand's Postulate, which guarantees a prime between any number (n) and (2n).
Starting from the fact that all integers from 7 to 19 can be written as sums of distinct primes from (${2,3,5,7,11}$), the proof uses induction and the inequality $(p_{i+1}<2p_i)$ to show that increasingly larger intervals of integers can also be represented as sums of distinct primes. By extending these intervals indefinitely, it follows that every integer ($n\ge 7$) has such a representation, illustrating a beautiful connection between prime distribution and additive number theory.
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Starting from the fact that all integers from 7 to 19 can be written as sums of distinct primes from (${2,3,5,7,11}$), the proof uses induction and the inequality $(p_{i+1}<2p_i)$ to show that increasingly larger intervals of integers can also be represented as sums of distinct primes. By extending these intervals indefinitely, it follows that every integer ($n\ge 7$) has such a representation, illustrating a beautiful connection between prime distribution and additive number theory.
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