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Sidst opdateret d. 22-04-2010 |
Perfekte tal |
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Et perfekt tal defineres
således:
Et tal, der er lig summen af alle de hele tal (1 inklusive), der går op i tallet, f.eks. 1 + 2 + 4 + 7 + 14 = 28. Euklid beviste, at udtrykket 2n¯¹ (2n-1) altid giver et fuldkomment tal, hvis udtrykket i parentes er et primtal, og det kræver, at n også er et primtal. At n er et primtal, medfører dog ikke altid, at parentesen er et primtal. Euler har vist, at formlen også giver alle lige fuldkomne tal; indtil nu har man ikke kendskab til ulige fuldkomne tal. De første 24 fuldkomne tal er beregnet (1977); det største af dem har n = 19.937 og har 12.003 cifre (siden er flere blevet beregnet - senest det 38. i 1999 med n = 6.972.593). Alle kendte fuldkomne tal ender på 6 eller 8, og for alle fuldkomne tal gælder, at summen af de reciprokke værdier af de tal, der går op i dem, er lig med 2, f.eks. 1/1 + 1/2 + 1/4 + 1/7 +1/14 + 1/28 = 2. |
6 med formlen (2^(2¯¹))*((2^2)-1) og divisorerne (1, 2, 3)
28 med formlen (2^(3¯¹))*((2^3)-1) og divisorerne (1, 2, 4, 7, 14)
496 med formlen (2^(5¯¹))*((2^5)-1) og divisorerne (1, 2, 4, 8, 16, 31, 62, 124, 248)
8128 med formlen (2^(7¯¹))*((2^7)-1) og divisorerne (1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064)
33.550.336 med formlen (2^(13¯¹))*((2^13)-1) og divisorerne (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8191, 16.382, 32.764, 65.528, 131.056, 262.112, 524.224, 1.048.448, 2.096.896, 4.193.792, 8.387.584, 16.775.168)
8.589.869.056 med formlen (2^(17¯¹))*((2^17)-1) og divisorerne (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16.384, 32.768, 65.536, 131.071, 262.142, 524.284, 1.048.568, 2.097.136, 4.194.272, 8.388.544, 16.777.088, 33.554.176, 67.108.352, 134.216.704, 268.433.408, 536.866.816, 1.073.733.632, 2.147.467.264, 4.294.934.528)
137.438.691.328 med formlen (2^(19¯¹))*((2^19)-1) og divisorerne (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16.384, 32.768, 65.536, 131.072, 262.144, 524.287, 1.048.574, 2.097.148, 4.194.296, 8.388.592, 16.777.184, 33.554.368, 67.108.736, 134.217.472, 268.434.944, 536.869.888, 1.073.739.776, 2.147.479.552, 4.294.959.104, 8.589.918.208, 17.179.836.416, 34.359.672.832, 68.719.345.664)
2.305.843.008.139.952.128 med formlen (2^(31¯¹))*((2^31)-1) og divisorerne (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16.384, 32.768, 65.536, 131.072, 262.144, 524.288, 1.048.576, 2.097.152, 4.194.304, 8.388.608, 16.777.216, 33.554.432, 67.108.864, 134.217.728, 268.435.456, 536.870.912, 1.073.741.824, 2.147.483.647, 4.294.967.294, 8.589.934.588, 17.179.869.176, 34.359.738.352, 68.719.476.704, 137.438.953.408, 274.877.906.816, 549.755.813.632, 1.099.511.627.264, 2.199.023.254.528, 4.398.046.509.056, 8.796.093.018.112, 17.592.186.036.224, 35.184.372.072.448, 70.368.744.144.896, 140.737.488.289.792, 281.474.976.579.584, 562.949.953.159.168, 1.125.899.906.318.336, 2.251.799.812.636.672, 4.503.599.625.273.344, 9.007.199.250.546.668, 18.014.398.501.093.376, 36.028.797.002.186.752, 72.057.594.004.373.504, 144.115.188.008.747.008, 288.230.376.017.494.016, 576.460.752.034.988.032, 1.152.921.504.069.976.064)
2.658.455.991.569.831.744.654.692.615.953.842.176 med formlen (2^(61¯¹))*((2^61)-1) og divisorerne (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16.384, 32.768, 65.536, 131.072, 262.144, 524.288, 1.048.576, 2.097.152, 4.194.304, 8.388.608, 16.777.216, 33.554.432, 67.108.864, 134.217.728, 268.435.456, 536.870.912, 1.073.741.824, 2.147.483.648, 4.294.967.296, 8.589.934.592, 17.179.869.184, 34.359.738.368, 68.719.476.736, 137.438.953.472, 274.877.906.944, 549.755.813.888, 1.099.511.627.776, 2.199.023.255.552, 4.398.046.511.104, 8.796.093.022.208, 17.592.186.044.416, 35.184.372.088.832, 70.368.744.177.664, 140.737.488.355.328, 281.474.976.710.656, 562.949.953.421.312, 1.125.899.906.842.624, 2.251.799.813.685.248, 4.503.599.627.370.496, 9.007.199.254.740.992, 18.014.398.509.481.984, 36.028.797.018.963.968, 72.057.594.037.927.936, 144.115.188.075.855.872, 288.230.376.151.711.744, 576.460.752.303.423.488, 1.152.921.504.606.846.976, 2.305.843.009.213.693.951, 4.611.686.018.427.387.902, 9.223.372.036.854.775.804, 18.446.744.073.709.551.608, 36.893.488.147.419.103.216, 73.786.976.294.838.206.432, 147.573.952.589.676.412.864, 295.147.905.179.352.825.728, 590.295.810.358.705.651.456, 1.180.591.620.717.411.302.912, 2.361.183.241.434.822.605.824, 4.722.366.482.869.645.211.648, 9.444.732.965.739.290.423.296, 18.889.465.931.478.580.846.592, 37.778.931.862.957.161.693.184, 75.557.863.725.914.323.386.368, 151.115.727.451.828.646.772.736, 302.231.454.903.657.293.545.472, 604.462.909.807.314.587.090.944, 1.208.925.819.614.629.174.181.888, 2.417.851.639.229.258.348.363.776, 4.835.703.278.458.516.696.727.552, 9.671.406.556.917.033.393.455.104, 19.342.813.113.834.066.786.910.208, 38.685.626.227.668.133.573.820.416, 77.371.252.455.336.267.147.640.832, 154.742.504.910.672.534.295.281.664, 309.485.009.821.345.068.590.563.328, 618.970.019.642.690.137.181.126.656, 1.237.940.039.285.380.274.362.253.312, 2.475.880.078.570.760.548.724.506.624, 4.951.760.157.141.521.097.449.013.248, 9.903.520.314.283.042.194.898.026.496, 19.807.040.628.566.084.389.796.052.992, 39.614.081.257.132.168.779.592.105.984, 79.228.162.514.264.337.559.184.211.968, 158.456.325.028.528.675.118.368.423.936, 316.912.650.057.057.350.236.736.847.872, 633.825.300.114.114.700.473.473.695.744, 1.267.650.600.228.229.400.946.947.391.488, 2.535.301.200.456.458.801.893.894.782.976, 5.070.602.400.912.917.603.787.789.565.952, 10.141.204.801.825.835.207.575.579.131.904, 20.282.409.603.651.670.415.151.158.263.808, 40.564.819.207.303.340.830.302.316.527.616, 81.129.638.414.606.681.660.604.633.055.232, 162.259.276.829.213.363.321.209.266.110.464, 324.518.553.658.426.726.642.418.532.220.928, 649.037.107.316.853.453.284.837.064.441.856, 1.298.074.214.633.706.906.569.674.128.883.712, 2.596.148.429.267.413.813.139.348.257.767.424, 5.192.296.858.534.827.626.278.696.515.534.848, 10.384.593.717.069.655.252.557.393.031.069.696, 20.769.187.434.139.310.505.114.786.062.139.392, 41.538.374.868.278.621.010.229.572.124.278.784, 83.076.749.736.557.242.020.459.144.248.557.568, 166.153.499.473.114.484.040.918.288.497.115.136, 332.306.998.946.228.968.081.836.576.994.230.272, 664.613.997.892.457.936.163.673.153.988.460.544, 1.329.227.995.784.915.872.327.346.307.976.921.088)
191.561.942.608.236.107.294.793.378.084.303.638.130.997.321.548.169.216 med formlen (2^(89¯¹))*((2^89)-1) og divisorerne (1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16.384, 32.768, 65.536, 131.072, 262.144, 524.288, 1.048.576, 2.097.152, 4.194.304, 8.388.608, 16.777.216, 33.554.432, 67.108.864, 134.217.728, 268.435.456, 536.870.912, 1.073.741.824, 2.147.483.648, 4.294.967.296, 8.589.934.592, 17.179.869.184, 34.359.738.368, 68.719.476.736, 137.438.953.472, 274.877.906.944, 549.755.813.888, 1.099.511.627.776, 2.199.023.255.552, 4.398.046.511.104, 8.796.093.022.208, 17.592.186.044.416, 35.184.372.088.832, 70.368.744.177.664, 140.737.488.355.328, 281.474.976.710.656, 562.949.953.421.312, 1.125.899.906.842.624, 2.251.799.813.685.248, 4.503.599.627.370.496, 9.007.199.254.740.992, 18.014.398.509.481.984, 36.028.797.018.963.968, 72.057.594.037.927.936, 144.115.188.075.855.872, 288.230.376.151.711.744, 576.460.752.303.423.488, 1.152.921.504.606.846.976, 2.305.843.009.213.693.952, 4.611.686.018.427.387.904, 9.223.372.036.854.775.808, 18.446.744.073.709.551.616, 36.893.488.147.419.103.232, 73.786.976.294.838.206.464, 147.573.952.589.676.412.928, 295.147.905.179.352.825.856, 590.295.810.358.705.651.712, 1.180.591.620.717.411.303.424, 2.361.183.241.434.822.606.848, 4.722.366.482.869.645.213.696, 9.444.732.965.739.290.427.392, 18.889.465.931.478.580.854.784, 37.778.931.862.957.161.709.568, 75.557.863.725.914.323.419.136, 151.115.727.451.828.646.838.272, 302.231.454.903.657.293.676.544, 604.462.909.807.314.587.353.088, 1.208.925.819.614.629.174.706.176, 2.417.851.639.229.258.349.412.352, 4.835.703.278.458.516.698.824.704, 9.671.406.556.917.033.397.649.408, 19.342.813.113.834.066.795.298.816, 38.685.626.227.668.133.590.597.632, 77.371.252.455.336.267.181.195.264, 154.742.504.910.672.534.362.390.528, 309.485.009.821.345.068.724.781.056, 618.970.019.642.690.137.449.562.112, 1.237.940.039.285.380.274.899.124.222, 2.475.880.078.570.760.549.798.248.444, 4.951.760.157.141.521.099.596.496.888, 9.903.520.314.283.042.199.192.993.776, 19.807.040.628.566.084.398.385.987.552, 39.614.081.257.132.168.796.771.975.104, 79.228.162.514.264.337.593.543.950.208, 158.456.325.028.528.675.187.087.900.416, 316.912.650.057.057.350.374.175.800.832, 633.825.300.114.114.700.748.351.601.664, 1.267.650.600.228.229.401.496.703.203.328, 2.535.301.200.456.458.802.993.406.406.656, 5.070.602.400.912.917.605.986.812.813.312, 10.141.204.801.825.835.211.973.625.626.624, 20.282.409.603.651.670.423.947.251.253.248, 40.564.819.207.303.340.847.894.502.506.496, 81.129.638.414.606.681.695.789.005.012.992, 162.259.276.829.213.363.391.578.010.025.984, 324.518.553.658.426.726.783.156.020.051.968, 649.037.107.316.853.453.566.312.040.103.936, 1.298.074.214.633.706.907.132.624.080.207.872, 2.596.148.429.267.413.814.265.248.160.415.744, 5.192.296.858.534.827.628.530.496.320.831.488, 10.384.593.717.069.655.257.060.992.641.662.976, 20.769.187.434.139.310.514.121.985.283.325.952, 41.538.374.868.278.621.028.243.970.566.651.904, 83.076.749.736.557.242.056.487.941.133.303.808, 166.153.499.473.114.484.112.975.882.266.607.616, 332.306.998.946.228.968.225.951.764.533.215.232, 664.613.997.892.457.936.451.903.529.066.430.464, 1.329.227.995.784.915.872.903.807.058.132.860.928, 2.658.455.991.569.831.745.807.614.116.265.721.856, 5.316.911.983.139.663.491.615.228.232.531.443.712, 10.633.823.966.279.326.983.230.456.465.062.887.424, 21.267.647.932.558.653.966.460.912.930.125.774.848, 42.535.295.865.117.307.932.921.825.860.251.549.696, 85.070.591.730.234.615.865.843.651.720.503.099.392, 170.141.183.460.469.231.731.687.303.441.006.198.784, 340.282.366.920.938.463.463.374.606.882.012.397.568, 680.564.733.841.876.926.926.749.213.764.024.795.136, 1.361.129.467.683.753.853.853.498.427.528.049.590.272, 2.722.258.935.367.507.707.706.996.855.056.099.180.544, 5.444.517.870.735.015.415.413.993.710.112.198.361.088, 10.889.035.741.470.030.830.827.987.420.224.396.722.176, 21.778.071.482.940.061.661.655.974.840.448.793.444.352, 43.556.142.965.880.123.323.311.949.680.897.586.888.704, 87.112.285.931.760.246.646.623.899.361.795.173.777.408, 174.224.571.863.520.493.293.247.798.723.590.347.554.816, 348.449.143.727.040.986.586.495.597.447.180.695.109.632, 696.898.287.454.081.973.172.991.194.894.361.390.219.264, 1.393.796.574.908.163.946.345.982.389.788.722.780.438.528, 2.787.593.149.816.327.892.691.964.779.577.445.560.877.056, 5.575.186.299.632.655.785.383.929.559.154.891.121.754.112, 11.150.372.599.265.311.570.767.859.118.309.782.243.508.224, 22.300.745.198.530.623.141.535.718.236.619.564.487.016.448, 44.601.490.397.061.246.283.071.436.473.239.128.974.032.896, 89.202.980.794.122.492.566.142.872.946.478.257.948.065.792, 178.405.961.588.244.985.132.285.745.892.956.515.896.131.584, 356.811.923.176.489.970.264.571.491.785.913.031.792.263.168, 713.623.846.352.979.940.529.142.983.571.826.063.584.526.336, 1.427.247.692.705.959.881.058.285.967.143.652.127.169.052.672, 2.854.495.385.411.919.762.116.571.934.287.304.254.338.105.344, 5.708.990.770.823.839.524.233.143.868.574.608.508.676.210.688, 11.417.981.541.647.679.048.466.287.737.149.217.017.352.421.376, 22.835.963.083.295.358.096.932.575.474.298.434.034.704.842.752, 45.671.926.166.590.716.193.865.150.948.596.868.069.409.685.504, 91.343.852.333.181.432.387.730.301.897.193.736.138.819.371.008, 182.687.704.666.362.864.775.460.603.794.387.472.277.638.742.016, 365.375.409.332.725.729.550.921.207.588.774.944.555.277.484.032, 730.750.818.665.451.459.101.842.415.177.549.889.110.554.968.064, 1.461.501.637.330.902.918.203.684.830.355.099.778.221.109.936.128, 2.923.003.274.661.805.836.407.369.660.710.199.556.442.219.872.256, 5.846.006.549.323.611.672.814.739.321.420.399.112.884.439.744.512, 11.692.013.098.647.223.345.629.478.642.840.798.225.768.879.489.024, 23.384.026.197.294.446.691.258.957.285.681.596.451.537.758.978.048, 46.768.052.394.588.893.382.517.914.571.363.192.903.075.517.956.096, 93.536.104.789.177.786.765.035.829.142.726.385.806.151.035.912.192, 187.072.209.578.355.573.530.071.658.285.452.771.612.302.071.824.384, 374.144.419.156.711.147.060.143.316.570.905.543.224.604.143.648.768, 748.288.838.313.422.294.120.286.633.141.811.086.449.208.287.297.536, 1.496.577.676.626.844.588.240.573.266.283.622.172.898.416.574.595.072, 2.993.155.353.253.689.176.481.146.532.567.244.345.796.833.149.190.144, 5.986.310.706.507.378.352.962.293.065.134.488.691.593.666.298.380.288, 11.972.621.413.014.756.705.924.586.130.268.977.383.187.332.596.760.576, 23.945.242.826.029.513.411.849.172.260.537.954.766.374.665.193.521.152, 47.890.485.652.059.026.823.698.344.521.075.909.532.749.330.387.042.304, 95.780.971.304.118.053.647.396.689.042.151.819.065.498.660.774.084.608)
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