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Palindromic Sums of
Powers of Consecutive Primes
rood Sums of Squares rood Sums of Cubes rood Sums of Powers rood comments


Introduction
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Palindromic numbers are numbers which read the same from

 p_right left to right (forwards) as from the right to left (backwards) p_left
Here are a few random examples : 12921, 1960691, 389909983

[ May 27, 2001 ]
Hans Havermann, inspired by the topics of this page, programmed his 400 MHz iMac
using Mathematica 4 and got an impressive output after seven days of computation.
Palindromic Sums of Powers of Consecutive Primes
Here is his largest record

 510808567765808015 
which is the sum of the squares of the following 95 consecutive primes
733267332 + 733267692 + 733267872 + ... + 733283172 + 733283292 + 733283412

The exclusive Palindromic Prime number 373 can be expressed as a sequence of
Five Consecutive Primes in two ways :

373 = 67 + 71 + 73 + 79 + 83
373 = 3 2 + 5 2 + 7 2 + 11 2 + 13 2

Five Consecutive Primes starting from 7 are ideal candidates for creating two more palindromic numbers :

7 x 11 x 13 x 17 x 19 = 323323
7 2 + 11 2 + 13 2 + 17 2 + 19 2 = 989

Tables
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Index Nr Base Square Expression 
Palindromic Sums of Powers of Consecutive Primes Length 
   
Sums of Cubes of SEVEN Consecutive Primes
by Carlos Rivera (Dec 18, 1998).
1 433 + 473 + 533 + 593 + 613 + 673 + 713 
1.423.2417
   
Sums of Squares of SEVEN Consecutive Primes
[2] by Hans Havermann (May 27, 2001).
2 157.9992 + 158.0032 + 158.0092 + 158.0172 +
158.0292 + 158.0472 + 158.0712		 
174.803.308.47112
1 22 + 32 + 52 + 72 + 112 + 132 + 172 
The Number of the Beast !   6663
   
Sums of Squares of FIVE Consecutive Primes
2 72 + 112 + 132 + 172 + 192 
9893
1 32 + 52 + 72 + 112 + 132 
3733
   
Sums of Squares of FOUR Consecutive Primes
? ?2 + ?2 + ?2 + ?2 
??
   
Sums of Squares of THREE Consecutive Primes
1 7512 + 7572 + 7612 
1.716.1717
   
Sums of Squares of TWO Consecutive Primes
2 1.0512 + 1.0612 
2.230.3227
1 1.0192 + 1.0212 
Prime Curios!   2.080.8027
   
Sums of Twin Prime Pairs
by G. L. Honaker, Jr. (Jan 31, 1999) - [1-9].
by Carlos Rivera (Feb 1, 1999) - [10-102]
by Hans Havermann (May 27, 2001) - [103-166]
103-166 349.888.997 + 349.888.999 = 699.777.996
349.515.497 + 349.515.499 = 699.030.996
349.287.947 + 349.287.949 = 698.575.896
349.272.947 + 349.272.949 = 698.545.896
349.257.947 + 349.257.949 = 698.515.896
348.727.397 + 348.727.399 = 697.454.796
(Undulating !)   348.484.847 + 348.484.849 = 696.969.696
347.994.797 + 347.994.799 = 695.989.596
347.909.297 + 347.909.299 = 695.818.596
347.404.247 + 347.404.249 = 694.808.496
347.212.247 + 347.212.249 = 694.424.496
346.984.697 + 346.984.699 = 693.969.396
346.383.647 + 346.383.649 = 692.767.296
345.712.097 + 345.712.099 = 691.424.196
345.222.047 + 345.222.049 = 690.444.096
345.196.547 + 345.196.549 = 690.393.096
339.934.487 + 339.934.489 = 679.868.976
339.863.987 + 339.863.989 = 679.727.976
339.444.437 + 339.444.439 = 678.888.876
339.348.437 + 339.348.439 = 678.696.876
339.085.937 + 339.085.939 = 678.171.876
338.474.837 + 338.474.839 = 676.949.676
337.969.787 + 337.969.789 = 675.939.576
337.944.287 + 337.944.289 = 675.888.576
337.707.287 + 337.707.289 = 675.414.576
337.570.787 + 337.570.789 = 675.141.576
337.530.287 + 337.530.289 = 675.060.576
337.287.737 + 337.287.739 = 674.575.476
336.797.687 + 336.797.689 = 673.595.376
336.535.187 + 336.535.189 = 673.070.376
336.126.137 + 336.126.139 = 672.252.276
335.843.087 + 335.843.089 = 671.686.176
335.474.537 + 335.474.539 = 670.949.076
329.964.977 + 329.964.979 = 659.929.956
329.787.977 + 329.787.979 = 659.575.956
329.419.427 + 329.419.429 = 658.838.856
329.101.427 + 329.101.429 = 658.202.856
328.858.877 + 328.858.879 = 657.717.756
328.383.827 + 328.383.829 = 656.767.656
327.742.277 + 327.742.279 = 655.484.556
327.575.777 + 327.575.779 = 655.151.556
326.787.677 + 326.787.679 = 653.575.356
326.146.127 + 326.146.129 = 652.292.256
325.353.527 + 325.353.529 = 650.707.056
319.247.417 + 319.247.419 = 638.494.836
319.202.417 + 319.202.419 = 638.404.836
318.535.367 + 318.535.369 = 637.070.736
(Undulating !)   318.181.817 + 318.181.819 = 636.363.636
317.813.267 + 317.813.269 = 635.626.536
316.944.167 + 316.944.169 = 633.888.336
316.792.667 + 316.792.669 = 633.585.336
316.479.617 + 316.479.619 = 632.959.236
316.353.617 + 316.353.619 = 632.707.236
309.646.457 + 309.646.459 = 619.292.916
309.444.407 + 309.444.409 = 618.888.816
308.540.357 + 308.540.359 = 617.080.716
308.297.807 + 308.297.809 = 616.595.616
307.762.757 + 307.762.759 = 615.525.516
307.666.757 + 307.666.759 = 615.333.516
307.287.707 + 307.287.709 = 614.575.416
307.095.707 + 307.095.709 = 614.191.416
306.863.657 + 306.863.659 = 613.727.316
306.742.157 + 306.742.159 = 613.484.316
306.646.157 + 306.646.159 = 613.292.316
9
55-102 (The Beast !)   306.333.107 + 306.333.109 = 612.666.216
305.595.557 + 305.595.559 = 611.191.116
244.409.441 + 244.409.443 = 488.818.884
243.373.841 + 243.373.843 = 486.747.684
243.196.841 + 243.196.843 = 486.393.684
243.015.341 + 243.015.343 = 486.030.684
242.510.291 + 242.510.293 = 485.020.584
(Beautiful !)   242.242.241 + 242.242.243 = 484.484.484
241.545.191 + 241.545.193 = 483.090.384
241.515.191 + 241.515.193 = 483.030.384
241.272.641 + 241.272.643 = 482.545.284
241.095.641 + 241.095.643 = 482.191.284
240.616.091 + 240.616.093 = 481.232.184
233.565.881 + 233.565.883 = 467.131.764
233.404.331 + 233.404.333 = 466.808.664
233.227.331 + 233.227.333 = 466.454.664
232.383.731 + 232.383.733 = 464.767.464
232.176.731 + 232.176.733 = 464.353.464
231.671.681 + 231.671.683 = 463.343.364
230.732.081 + 230.732.083 = 461.464.164
230.702.081 + 230.702.083 = 461.404.164
230.308.031 + 230.308.033 = 460.616.064
224.979.971 + 224.979.973 = 449.959.944
224.636.471 + 224.636.473 = 449.272.944
223.651.871 + 223.651.873 = 447.303.744
223.025.321 + 223.025.323 = 446.050.644
222.560.771 + 222.560.773 = 445.121.544
222.505.271 + 222.505.273 = 445.010.544
222.348.221 + 222.348.223 = 444.696.444
221.252.621 + 221.252.623 = 442.505.244
220.626.071 + 220.626.073 = 441.252.144
220.287.521 + 220.287.523 = 440.575.044
214.202.411 + 214.202.413 = 428.404.824
213.863.861 + 213.863.863 = 427.727.724
212.419.211 + 212.419.213 = 424.838.424
211.641.161 + 211.641.163 = 423.282.324
211.232.111 + 211.232.113 = 422.464.224
210.893.561 + 210.893.563 = 421.787.124
210.348.011 + 210.348.013 = 420.696.024
204.974.951 + 204.974.953 = 409.949.904
202.944.251 + 202.944.253 = 405.888.504
202.439.201 + 202.439.203 = 404.878.404
202.232.201 + 202.232.203 = 404.464.404
201.575.651 + 201.575.653 = 403.151.304
201.499.601 + 201.499.603 = 402.999.204
201.484.601 + 201.484.603 = 402.969.204
200.828.051 + 200.828.053 = 401.656.104
200.323.001 + 200.323.003 = 400.646.004
9
34-54 34.894.397 + 34.894.399 = 69.788.796
34.711.247 + 34.711.249 = 69.422.496
33.633.137 + 33.633.139 = 67.266.276
33.600.137 + 33.600.139 = 67.200.276
32.972.477 + 32.972.479 = 65.944.956
32.522.027 + 32.522.029 = 65.044.056
30.677.657 + 30.677.659 = 61.355.316
30.661.157 + 30.661.159 = 61.322.316
30.600.107 + 30.600.109 = 61.200.216
24.483.491 + 24.483.493 = 48.966.984
24.227.741 + 24.227.743 = 48.455.484
23.494.481 + 23.494.483 = 46.988.964
23.477.981 + 23.477.983 = 46.955.964
23.222.231 + 23.222.233 = 46.444.464
23.194.181 + 23.194.183 = 46.388.364
22.366.871 + 22.366.873 = 44.733.744
22.005.521 + 22.005.523 = 44.011.044
21.422.411 + 21.422.413 = 42.844.824
20.461.451 + 20.461.453 = 40.922.904
20.283.251 + 20.283.253 = 40.566.504
20.161.151 + 20.161.153 = 40.322.304
8
13-33 3.450.047 + 3.450.049 = 6.900.096
3.375.287 + 3.375.289 = 6.750.576
3.296.477 + 3.296.479 = 6.592.956
3.289.877 + 3.289.879 = 6.579.756
3.270.227 + 3.270.229 = 6.540.456
3.263.627 + 3.263.629 = 6.527.256
3.185.867 + 3.185.869 = 6.371.736
3.087.857 + 3.087.859 = 6.175.716
(Undulating !)   3.080.807 + 3.080.809 = 6.161.616
2.442.941 + 2.442.943 = 4.885.884
2.434.841 + 2.434.843 = 4.869.684
2.431.841 + 2.431.843 = 4.863.684
2.422.241 + 2.422.243 = 4.844.484
2.404.541 + 2.404.543 = 4.809.084
2.328.281 + 2.328.283 = 4.656.564
2.303.531 + 2.303.533 = 4.607.064
2.217.671 + 2.217.673 = 4.435.344
2.129.261 + 2.129.263 = 4.258.524
2.039.351 + 2.039.353 = 4.078.704
2.037.851 + 2.037.853 = 4.075.704
2.025.251 + 2.025.253 = 4.050.504
7
10-12 202.751 + 202.753 = 405.504
211.661 + 211.663 = 423.324
240.041 + 240.043 = 480.084
6
5-9 (Palindromic Square !) (Undulating !)   34.847 + 34.849 = 69.696
33.587 + 33.589 = 67.176
30.557 + 30.559 = 61.116
22.271 + 22.273 = 44.544
21.011 + 21.013 = 42.024
5
3-4 3.167 + 3.169 = 6.336
2.111 + 2.113 = 4.224
4
2 (Undulating !)   347 + 349 = 696
3
1 3 + 5 = 8
1


Sources Revealed
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Hans Havermann, Palindromic Sums of Powers of Consecutive Primes, [ May 27, 2001 ].

Contributions
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Hans Havermann
Carlos Rivera
G. L. Honaker, Jr.

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(All rights reserved) - Last modified : March 29, 2007.
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E-mail address : pdg@worldofnumbers.com