250 Problems in Elementary Number Theory - Sierpinski (1970)

PROBLEMS 777. Prove that every prime of the form 4k + 1 is a hypotenuse of a rectangulartriangle with integer sides.78. Find four solutions of the equation p2+ 1 = q2+r2 with primes p,q, and r.79. Prove that the equation p2+q2 = r2+s2+t2 has no solution withprimesp, q, r, s, t.80*. Find all prime solutions p, q, r of the equation p(p+l)+q(q+l)= r(r+l).81*. Find all primesp, q, and r such that the numbers p(p+l), q(q+l),r(r+ 1) form an increasing arithmetic progression.82. Find all positive integers n such that each of the numbers n+ 1,n+3, n+7, n+9, n+13, and n+15 is a prime.83. Find five primes which are sums of two fourth powers of integers.84. Prove that there exist infinitely many pairs of consecutive primeswhich are not twin primes.85. Using the theorem of Lejeune-Dirichlet on arithmetic progressions,prove that there exist infinitely many primes which do not belong to anypair of twin primes.86. Find five least positive integers for which n 2 -1 is a product ofthree different primes.87. Find five least positive integers n for which n 2 + 1 is a product ofthree different primes, and find a positive integer n for which n 2 + 1 is aproduct of three different odd p~mes.88*. Prove that among each three consecutive integers > 7 at leastone has at least two different prime divisors.89. Find five least positive integers n such that each of the numbers n,n+ 1, n+2 is a product of two different primes. Prove that there are nofour consecutive positive integers with this property. Show by an examplethat there exist four positive integers such that each of them has exactlytwo different prime divisors.90. Prove that the theorem asserting that there exist only finitely manypositive integers n such that both nand n+ 1 have only one prime divisoris equivalent to the theorem asserting that there exist only finitely manyprime Mersenne numbers and finitely many prime Fermat numbers.