Second Prime Number Puzzle

  

Find the smallest number of prime addends that add up to 135, 555, 777, etc.

This page relates to finding ANY SUM that is in this list:
101 107 113 117 119 121 123 125 127 131 135 137 143 145 147 149 155 157 161 163 167 171 173 177 179 185 187 189 191 197 203 205 207 209 211 215 217 219 221 223 227 233 237 239 245 247 249 251 255 257 261 263 267 269 275 277 281 287 289 291 293 297 299 and any number listed in the full list, which goes from 23 to 997, further down.

This page is harder than the prior puzzle. However, to make it a little easier for you, here, again, is the list of all the prime numbers up to 1000. After you've found your own method, scrol down and I'll give you a few hints, though not a surefire strategy.

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
53 59 61 67 71 73 79 83 89 97 101 103 107
109 113 127 131 137 139 149 151 157 163 167
173 179 181 191 193 197 199 211 223 227 229
233 239 241 251 257 263 269 271 277 281 283
293 307 311 313 317 331 337 347 349 353 359
367 373 379 383 389 397 401 409 419 421 431
433 439 443 449 457 461 463 467 479 487 491
499 503 509 521 523 541 547 557 563 569 571
577 587 593 599 601 607 613 617 619 631 641
643 647 653 659 661 673 677 683 691 701 709
719 727 733 739 743 751 757 761 769 773 787
797 809 811 821 823 827 829 839 853 857 859
863 877 881 883 887 907 911 919 929 937 941
947 953 967 971 977 983 991 997

As you already figured out, none of these sums has a "two prime number" solution. In order to have a "prime pair" for the answer, one of the numbers would have to be 2, the only even prime number.

If your teacher or a friend gives you an even number greater than 6, and asks you to find two or more prime numbers that add up to that number, you will ALWAYS be able to find a pair of numbers to fit the bill. That was the topic of the prior page in this series.

If your teacher or friend gives you an odd number greater than 22, and asks you to find two or more different prime numbers that add up to that number, you should immediately check to see if the number (two less than the sum) is a prime number. If so, the answer will be the pair of numbers, (two) and (two less than the sum). If the number (two less than the sum) is not a prime number, then you will have to look for a "prime triple" for an answer. Every odd number greater than 22 has either a double or a triple answer, sometimes both, that can be found.

Related question: What is the largest sum that cannot be made by adding two or more positive prime numbers. I imply 22 above, but the answer is less than 22.

The strategy to find an answer is more involved when the sum is an odd number.

• Check the number (two less than the suggested sum)
• Is that number prime.
• If so, your answer is a "prime pair", (two) and (that other number)
  
  If (two less than the sum) is not prime, you have a harder job ahead of you.
• The answer will be a "prime triple", three odd prime numbers.
• Start with the lowest odd prime number.
• Subtract that number from your target sum. This is your "new sum"
• Use the technique for the prior puzzle to find two numbers that add up to that "new sum"
• If two numbers add up to that sum, you're done.
• If you don't find two numbers that add up to that number, try the next higher prime number for your first number.
• Remember, if you are working on 5 as your first number, you don't need to check for "prime pairs" with 3 or 5 in them, because you already checked 3 as the first number and didn't find one, and since the puzzle requires three different numbers, you don't check 5 again.
• I realize this is confusing, but it's 2 in the morning as I write this, and I'm in a hurry to go home. I've been working on these pages for eight hours now.
• Remember, you can use ANY of the target sums listed below. I've checked, and all of these sums have three odd prime numbers that add up to these sums. None of these numbers have a pair that add up to that number.

You will need to find three different positive prime numbers in order to get these harder sums. 23 27 29 35 37 41 47 51 53 57 59 65 67 71 77 79 83 87 89 93 95 97 101 107 113 117 119 121 123 125 127 131 135 137 143 145 147 149 155 157 161 163 167 171 173 177 179 185 187 189 191 197 203 205 207 209 211 215 217 219 221 223 227 233 237 239 245 247 249 251 255 257 261 263 267 269 275 277 281 287 289 291 293 297 299 301 303 305 307 311 317 321 323 325 327 329 331 335 337 341 343 345 347 353 357 359 363 365 367 371 373 377 379 383 387 389 393 395 397 401 405 407 409 413 415 417 419 425 427 429 431 437 439 443 447 449 453 455 457 461 467 471 473 475 477 479 483 485 487 491 495 497 499 503 507 509 513 515 517 519 521 527 529 531 533 535 537 539 541 545 547 551 553 555 557 561 563 567 569 575 577 581 583 585 587 591 593 597 599 605 607 611 613 617 623 625 627 629 631 635 637 639 641 647 651 653 657 659 665 667 669 671 673 677 681 683 687 689 691 695 697 699 701 705 707 709 713 715 717 719 723 725 727 731 733 737 739 743 747 749 751 755 757 761 765 767 769 773 777 779 781 783 785 787 791 793 795 797 801 803 805 807 809 815 817 819 821 827 833 835 837 839 843 845 847 849 851 853 857 863 867 869 871 873 875 877 881 887 891 893 895 897 899 901 903 905 907 911 915 917 919 923 925 927 929 933 935 937 941 945 947 951 953 957 959 961 963 965 967 971 975 977 981 983 987 989 991 995 997 (you will only need two different prime numbers to add up to any number greater than 23 not in this list)

Here's a related question again?

11 cannot be found as a sum of two or more prime numbers. What is the largest number that cannot be made as the sum of two or more positive prime numbers?

Have you figured out the answer yet. I'm not going to answer it for you anywhere on these pages.

When you are ready, we can look at the third puzzle.

Find the largest number of prime addends that add up to 1000

As you solve this puzzle, you only need the first few prime numbers, maybee
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97