What needs to be added to the sum of the first three prime numbers to equal the fifth?
Divergence of the sum of the reciprocals of the primes these primes. Then each of these primes divides all but one of the numerator terms and hence does not divide the numerator itself; but each prime "does" divide the denominator. Thus the expression is irreducible and is non-integer. Divergence of the sum of the reciprocals of the primes The sum of the reciprocals of all prime numbers diverges; that is: This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers. There are a variety of proofs of Euler's result, including a lower bound for the partial sums stating that for
To the Memory of Mr. Oldham slippery place, While his young friend performed and won the race. O early ripe! to thy abundant store What could advancing age have added more? It might (what nature never gives the young) Have taught the numbers of thy native tongue. But satire needs not those, and wit will shine Through the harsh cadence of a rugged line. A noble error, and but seldom made, When poets are by too much force betrayed. Thy generous fruits, though gathered ere their prime, Still showed a quickness; and maturing time But mellows what we write to the dull sweets of rhyme. Once