Mathematical Gems, Issue 2Mathematical Association of America, 1976 - Combinatorial analysis |
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Page 33
... implying that | B2 - 1 B2 | B2 1 | B21 . 1. This makes f ( x , y ) = 2 , a prime . ( b ) B2 = 0 : For B2 = 0 the value of the function is f ( x , y ) = = y 2 y 2 1 - [ I − 1 − ( − 1 ) ] + 2 - ' z 1 [ 1 + 1 ] + 2 = y − 1 + 2 = y + 1 ...
... implying that | B2 - 1 B2 | B2 1 | B21 . 1. This makes f ( x , y ) = 2 , a prime . ( b ) B2 = 0 : For B2 = 0 the value of the function is f ( x , y ) = = y 2 y 2 1 - [ I − 1 − ( − 1 ) ] + 2 - ' z 1 [ 1 + 1 ] + 2 = y − 1 + 2 = y + 1 ...
Page 164
... implying that p is at least 12. If none of P1 , P2 , P3 , is 3 , then each is congruent ( mod 3 ) to ± 1 . Thus 2 2 2 P = Pi + P2 + P3 = 1 + 1 + 1 = 0 ( mod 3 ) , contradicting the primality of p since p is at least 12. Thus at least ...
... implying that p is at least 12. If none of P1 , P2 , P3 , is 3 , then each is congruent ( mod 3 ) to ± 1 . Thus 2 2 2 P = Pi + P2 + P3 = 1 + 1 + 1 = 0 ( mod 3 ) , contradicting the primality of p since p is at least 12. Thus at least ...
Page 169
... implying that z cannot go into x + y more often than once . Thus z = x + y . This makes x + z = 2x + y , and this is divisible by y . Con- sequently , y must divide 2x . However , 2x < 2y , implying that y could not go into 2x more than ...
... implying that z cannot go into x + y more often than once . Thus z = x + y . This makes x + z = 2x + y , and this is divisible by y . Con- sequently , y must divide 2x . However , 2x < 2y , implying that y could not go into 2x more than ...
Contents
CHAPTER PAGE 1 Three Surprises from Combinatorics and Number Theory | 1 |
Four Minor Gems from Geometry | 10 |
A Problem in CheckerJumping | 23 |
Copyright | |
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1-factor a₁ a₂ Accordingly Amer attainable score beads BICENTRIC POLYGONS black edges circumcenter circumcircle collinear column complete graph configuration congruent Consequently contains contradiction copies deleted diagonal digits distance can occur divides dominoes eight-point circle equal exceed Figure Gabriel Lamé gives graph G harmonic bricks harmonic series hexominoes implying incircle integer intersection isosceles 6-point isosceles tetrahedron key multiple Klarner L-tromino lattice point layers least line ax Math monomino Monthly n-gon natural numbers necklaces number of vertices obtain odd block odd components odd number odd prime P₁ pack the line packable pair Paul Erdös plane polyominoes prime number problem proof proved quadrilateral rectangle red edges relatively prime result segments sides Similarly Slothouber-Graatsma Puzzle spheres subset Suppose tangent tetrahedron tetrominoes theorem total number tromino unattainable unit cubes vertex Wilson's theorem yields