Transition Year Maths
Fermat loved mathematical challenges. He noted that 26 is between 25 (which is 52
) and 27 (which is 33
). See if you can
find any other number sandwiched between a square and a cube? You will not find any. Fermat proved that no other number could have this property. This proof was just one of hundreds of proofs he completed.
PROJECT 12.4 Equations: Power of 4 Write out what 14
, 24 , 34 , 44 + y4 + z4 and 54 are.
Then try all the combinations to verify that none solve the equation, x4
= w4
PROJECT 12.5 Equations: Whole Numbers Write out a value for 14
, 24 84 , 94 and 104 + y4 , 34 = z4 , 44 + w4 , 54 , 64 + 32
Christian Goldbach: Prussian mathematician, (1690–1764)
, 74 , . Then using a different whole
number for x, y, z and w between 1 and 10 find a solution to x4
Euler’s Conjecture
There was another conjecture called Euler’s Conjecture which stated that there were no solutions to the equation: x4
+ y4 + z4 = w4
Euler’s Conjecture was never proved, but was assumed to be true because so many people tried so many numbers without success. Even with years of computer-sifting the lack of a counter example made it likely this conjecture was true.
Then amazingly a man called Naom Elkies in 1988 found 2,682,4404
+ 15,365,6394 + 18,796,7604 = 20,615,6734 You can check this to 9 places of decimals on your calculator.
Goldbach was born in Kaliningrad, Russia. He worked closely with Leibniz and Euler on number theory and infinite series. He is most famous for his theory as discussed below. British publishers Faber & Faber offered $1,000,000 if anyone could prove the conjecture before April 2002, but the deadline passed and no one came forward with a proof. In a Spanish movie called La Habitacion de Fermat (2007) a young mathematician claims to have solved it. In real life the conjecture remains unsolved.
Goldbach’s Conjecture
Goldbach’s Conjecture was proposed by Christian Goldbach (1690 – 1764). When Andrew Wiles was in Dublin in 2003 he was asked if he was now working on Goldbach’s Conjecture – he did not reply. The conjecture states, ‘Every even number greater than 2 is the sum of two prime numbers’. Pick any even number. 20 can be written as the sum of 13 and 7; two prime numbers.
116
Page 1 |
Page 2 |
Page 3 |
Page 4 |
Page 5 |
Page 6 |
Page 7 |
Page 8 |
Page 9 |
Page 10 |
Page 11 |
Page 12 |
Page 13 |
Page 14 |
Page 15 |
Page 16 |
Page 17 |
Page 18 |
Page 19 |
Page 20 |
Page 21 |
Page 22 |
Page 23 |
Page 24 |
Page 25 |
Page 26 |
Page 27 |
Page 28 |
Page 29 |
Page 30 |
Page 31 |
Page 32 |
Page 33 |
Page 34 |
Page 35 |
Page 36 |
Page 37 |
Page 38 |
Page 39 |
Page 40 |
Page 41 |
Page 42 |
Page 43 |
Page 44 |
Page 45 |
Page 46 |
Page 47 |
Page 48 |
Page 49 |
Page 50 |
Page 51 |
Page 52 |
Page 53 |
Page 54 |
Page 55 |
Page 56 |
Page 57 |
Page 58 |
Page 59 |
Page 60 |
Page 61 |
Page 62 |
Page 63 |
Page 64 |
Page 65 |
Page 66 |
Page 67 |
Page 68 |
Page 69 |
Page 70 |
Page 71 |
Page 72 |
Page 73 |
Page 74 |
Page 75 |
Page 76 |
Page 77 |
Page 78 |
Page 79 |
Page 80 |
Page 81 |
Page 82 |
Page 83 |
Page 84 |
Page 85 |
Page 86 |
Page 87 |
Page 88 |
Page 89 |
Page 90 |
Page 91 |
Page 92 |
Page 93 |
Page 94 |
Page 95 |
Page 96 |
Page 97 |
Page 98 |
Page 99 |
Page 100 |
Page 101 |
Page 102 |
Page 103 |
Page 104 |
Page 105 |
Page 106 |
Page 107 |
Page 108 |
Page 109 |
Page 110 |
Page 111 |
Page 112 |
Page 113 |
Page 114 |
Page 115 |
Page 116 |
Page 117 |
Page 118 |
Page 119 |
Page 120 |
Page 121 |
Page 122 |
Page 123 |
Page 124 |
Page 125 |
Page 126 |
Page 127 |
Page 128 |
Page 129 |
Page 130 |
Page 131 |
Page 132 |
Page 133 |
Page 134 |
Page 135 |
Page 136 |
Page 137 |
Page 138 |
Page 139 |
Page 140 |
Page 141 |
Page 142 |
Page 143 |
Page 144 |
Page 145 |
Page 146 |
Page 147 |
Page 148 |
Page 149 |
Page 150 |
Page 151 |
Page 152 |
Page 153 |
Page 154 |
Page 155 |
Page 156 |
Page 157 |
Page 158 |
Page 159 |
Page 160 |
Page 161 |
Page 162 |
Page 163 |
Page 164 |
Page 165 |
Page 166 |
Page 167 |
Page 168
