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German

# Do primes arise from wrong binomic formulae?

When at school, we had to learn formulae called "binomische Formeln" in German; I do not know the English term for them. Search with Google led me to the binomial theorem which is very near, but more general (also for higher powers, not only for quadratic expressions), and which includes the binomial coefficients while the formulae we had to learn had only an explicit 2.

Perhaps you remember the three formulae

```          (a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
(a + b) (a - b) = a² - b²
```
The first two are very similar; you only have to remember that the minus sign is only with the middle term, precisely not with the b². The third formula was called the "mixed binomic forumla", and it has the minus with the quadratic term b². Its right side is simpler than the left, opposite to the first two formulae.

The difficulty with these formulae was not to use them left to right, but to use them backwards, right to left. As an example, we should simplify a fraction like

```          a² - b²
-------  , giving  1/3 (a + b)
3a - 3b
```

There seems no connection to prime numbers - or is there a hidden connection? One can test and insert some numbers for a and b. We limit the discussion to positive integers a and b, having a > b, in the following.

Searching for primes, it is not an good idea to start with an expression a² - b²; the mixed formula gives exactly a factorisation, and therefore, the result of a² - b² is not a prime, except for the special case a - b = 1. With this special case, one can reach all odd numbers, and therefore all primes but 2. This is noted on the margin only, here.

Numbers from the first two formulae are not suitable either; they are squares, and a square is not a prime.

It happened now and then that we computed errorneously (a + b)² = a² + b²; I call it error 1. The tentation was to do it like the multiplication rule (the distributive law) which states that (a + b) * 2 = a * 2 + b * 2.

Error 2 is to omit the coefficient 2 of the middle term, i.e. to 'compute' (a + b)² = a² + ab + b². And error 3 is to set the minus sign also with the b², i.e. to 'compute' (a - b)² = a² - ab - b². In the following text, I use this third term with a plus sign for the middle term; this is equivalent: if you can compute a number as a²-ab-b², then you can write it as a²+ab-b² also, for some smaller a and b.

Intresting (at least for me) is that these three error terms

```          a² + b²
a² + ab + b²
a² + ab - b²
```
yield rather many prime numbers, and that one can see some schema or some type of order with it. This is shown in the following table. Depending on the type, the primes are shown on different background.

Primes of form a²+b² can be factorised if you work with Gaussian integers. Every prime which gives a remainder of 1 when divided by 4 is of form a²+b², and is shown with yellow background.

For some primes, it happens that we get them by two (or even all tree? I do not know) different terms. For instance, one gets 13 both as 3² + 2² = 9 + 4 (error 1 term), and also as 3² + 3 * 1 + 1² (error 2 term).

Primes of form a²+b² have yellow background, not depending on if they allow another "error term" or not. Blue background stands for "only a² + ab + b²", red for "only a² + ab - b²", and green for "both second an third error term is possible".

The table is very economic, maybe it can be called spartanic. Numbers divisible by 2, 3 or 5 are completely omitted. For a range of 30 numbers, there are only 8 possibilities to find a prime. I started the table with these 8 colums. This gives a table which is almost bewildering by its colorfulness.

Then I realized that there is only one color for a column when I double the range of each line to 60 numbers, containing 16 places for primes. That is how you see the table now.

Beyond that: all prime numbers in a column are colored, and they all get the same color. Eight columns out of the 16 contain primes of form a²+b² (yellow), two primes of form a²+ab+b² (blue), two colums primes of form a²+ab-b² (red), and two columns contain primes which allow the second and the third "error fromula" (green). The last two two columns contain primes which do not fit to any of the three "error formulae".

One can even find the primes of form a² + b² in the OEIS (Online Encyclopedia of Integer Sequences), under A002144. Primes of form a²+ab+b² are under A068229, and those of form a²+ab-b² under A141302. And you find also the primes not fitting to any "error formula", under A107169.

 Legend Prime can be written as a²+b² Prime can be written as a²+ab+b² Prime can be written as a²+ab-b² Both a²+ab+b² and a²+ab-b² possible Not a prime, a product a*b Prime without simple binomic expression

 + 1 + 7 + 11 + 13 + 17 + 19 + 23 + 29 +31 +37 + 41 + 43 + 47 + 49 + 53 + 59 0 1 2_1 3_1 3_2 4_1 3_2 23 5_2 5_1 6_1 5_4 6_1 47 7 * 7 7_2 7_2 60 6_5 7_2 8_1 8_3 7 * 11 7_3 83 8_5 7 * 13 9_4 10_1 9_2 107 10_3 8_7 7 * 17 120 11 * 11 7_6 11_1 7 * 19 11_4 10_3 11 * 13 10_7 9_5 11_6 7 * 23 11_3 167 13 * 13 13_2 12_5 180 10_9 11 * 17 13_2 12_7 14_1 13_2 7 * 29 11 * 19 14_1 7 * 31 13 * 17 11_6 227 15_2 13_8 15_1 240 15_4 13 * 19 15_2 11 * 23 16_1 7 * 37 263 13_10 10_9 14_9 16_5 13_6 7 * 41 17 * 17 17_2 13 * 23 300 7 * 43 17_1 16_5 13_12 14_11 11 * 29 17 * 19 7 * 47 11_10 16_9 11 * 31 7 * 49 347 18_5 17_8 17_7 360 19 * 19 13_9 7 * 53 18_7 13 * 29 15_7 383 17_10 17 * 23 19_6 20_1 13 * 31 11 * 37 20_3 7 * 59 20_1 420 15_14 7 * 61 19_5 17_12 19 * 23 18_5 443 20_7 11 * 41 21_4 19_10 21_1 467 7 * 67 11 * 43 21_2 480 13 * 37 21_2 20_7 17 * 29 7 * 71 18_7 503 22_5 7 * 73 11 * 47 20_11 17_9 17 * 31 23 * 23 13 * 41 11 * 49 540 21_10 14_13 19 * 29 7 * 79 19_14 13 * 43 563 20_13 21_5 24_1 7 * 83 11 * 53 587 19 * 31 23_8 24_1 600 24_5 23_3 13 * 47 18_17 19_16 22_5 7 * 89 17 * 37 15_14 13 * 49 25_4 18_11 647 11 * 59 22_13 23_10 660 25_6 23 * 29 11 * 61 23_12 26_1 7 * 97 683 13 * 53 19_11 17 * 41 26_5 19 * 37 7 * 101 22_15 23 * 31 24_11 720 7 * 103 18_13 17 * 43 27_2 11 * 67 23_7 743 7 * 107 21_10 26_9 20_19 7 * 109 13 * 59 25_12 22_17 19 * 41 780 11 * 71 27_2 7 * 113 13 * 61 26_11 17 * 47 11 * 73 28_5 25_6 19 * 43 25_14 19_14 827 27_10 17 * 49 27_5 840 29 * 29 11 * 77 23 * 37 23_18 29_4 23_10 863 11 * 79 13 * 67 29_6 25_16 21_13 887 7 * 127 19 * 47 29 * 31 900 17 * 53 26_7 27_13 11 * 83 7 * 131 18_17 13 * 71 23_20 19 * 49 24_19 29_10 23 * 41 947 13 * 73 28_13 7 * 137 960 31 * 31 27_7 28_11 7 * 139 31_4 11 * 89 983 23 * 43 26_9 31_6 13 * 77 17 * 59 19 * 53 28_15 23_22 31_2
 + 1 + 7 + 11 + 13 + 17 + 19 + 23 + 29 +31 +37 + 41 + 43 + 47 + 49 + 53 + 59 1020 30_11 13 * 79 29_10 32_3 17 * 61 22_15 7 * 149 32_5 29_6 7 * 151 31_10 31_3 11 * 97 30_13 29 * 37 13 * 83 1080 23 * 47 21_17 31_5 33_2 29_16 7 * 157 1103 25_22 11 * 101 26_21 19 * 59 33_1 23 * 49 27_20 11 * 103 17 * 67 1140 7 * 163 31 * 37 33_2 33_8 13 * 89 19 * 61 1163 7 * 167 25_14 11 * 107 34_5 13 * 91 1187 29 * 41 32_13 11 * 109 1200 25_24 17 * 71 7 * 173 27_22 31_16 23 * 53 1223 35_2 29_10 34_9 17 * 73 11 * 113 29 * 43 32_15 7 * 179 35_1 1260 13 * 97 7 * 181 31 * 41 19 * 67 34_11 33_5 1283 35_8 26_15 36_1 26_25 27_14 1307 17 * 77 13 * 101 33_10 1320 36_5 23_19 11 * 121 31 * 43 7 * 191 13 * 103 17 * 79 19 * 71 7 * 193 23 * 59 31_20 29 * 47 1367 37 * 37 37_2 7 * 197 1380 34_15 19 * 73 13 * 107 7 * 199 11 * 127 25_18 23 * 61 28_25 17 * 83 13 * 109 29 * 49 31_11 1427 30_23 37_8 37_2 1440 11 * 131 37_2 36_5 38_3 31 * 47 30_13 19 * 77 13 * 113 35_6 7 * 211 35_16 38_1 1487 33_20 38_7 36_7 1500 19 * 79 11 * 137 35_13 17 * 89 37 * 41 31 * 49 1523 11 * 139 26_19 29 * 53 23 * 67 34_9 17 * 91 35_18 32_23 39_1 1560 7 * 223 38_3 36_11 13 * 121 19 * 83 37_5 1583 7 * 227 37 * 43 34_21 40_1 7 * 229 1607 40_3 38_13 36_17 1620 39_10 37_6 7 * 233 23 * 71 31_26 11 * 149 31 * 53 17 * 97 13 * 127 36_19 11 * 151 26_21 1667 38_15 7 * 239 23 * 73 1680 41 * 41 7 * 241 19 * 89 37_18 41_4 30_17 13 * 131 35_22 29 * 59 17 * 101 40_11 41_1 11 * 157 19 * 91 38_17 37 * 47 1740 30_29 33_14 17 * 103 32_27 7 * 251 38_7 41 * 43 29 * 61 23 * 77 39_16 13 * 137 37_9 1787 42_5 11 * 163 7 * 257 1800 35_24 13 * 139 39_10 37 * 49 23 * 79 17 * 107 1823 31 * 59 30_19 11 * 167 7 * 263 19 * 97 1847 43 * 43 17 * 109 13 * 143 1860 31_30 38_9 39_14 33_28 41_14 27_23 7 * 269 40_17 31 * 61 7 * 271 35_26 11 * 173 1907 23 * 83 43_8 19 * 101 1920 17 * 113 41 * 47 43_2 42_13 13 * 149 7 * 277 29 * 67 43_10 26_25 19 * 103 37 * 53 13 * 151 7 * 281 11 * 179 38_23 44_1
 + 1 + 7 + 11 + 13 + 17 + 19 + 23 + 29 +31 +37 + 41 + 43 + 47 + 49 + 53 + 59 1980 7 * 283 43_3 11 * 181 43_12 34_29 42_5 2003 41 * 49 39_10 44_9 43 * 47 17 * 119 2027 45_2 19 * 107 43_5 2040 13 * 157 23 * 89 7 * 293 42_17 17 * 121 29 * 71 2063 38_25 19 * 109 31 * 67 41_20 37_14 2087 45_8 23 * 91 41_19 2100 11 * 191 43 * 49 45_2 33_32 29 * 73 13 * 163 11 * 193 40_23 41_9 36_29 46_5 43_6 19 * 113 7 * 307 37_28 17 * 127 2160 44_15 11 * 197 13 * 167 41 * 53 7 * 311 35_18 37 * 59 11 * 199 7 * 313 13 * 169 31 * 71 31_23 2207 47 * 47 47_2 7 * 317 2220 45_14 17 * 131 23 * 97 29 * 77 46_11 38_15 2243 13 * 173 35_19 37 * 61 19 * 119 31 * 73 2267 37_30 47_8 43 * 53 2280 45_16 37_17 29 * 79 42_23 44_19 19 * 121 47 * 49 47_10 34_21 7 * 331 11 * 211 23 * 101 13 * 179 17 * 137 43_22 44_13 2340 46_15 42_11 48_1 13 * 181 41_26 7 * 337 17 * 139 23 * 103 46_5 44_21 35_34 41_13 31 * 77 42_25 37_32 45_11 2400 49 * 49 29 * 83 44_19 19 * 127 49_4 41 * 59 2423 7 * 347 17 * 143 49_6 40_29 7 * 349 2447 31 * 79 11 * 223 45_14 2460 23 * 107 34_23 7 * 353 48_13 46_19 37 * 67 13 * 191 19 * 131 47 * 53 11 * 227 41 * 61 49_2 23 * 109 13 * 193 7 * 359 11 * 229 2520 36_35 19 * 133 45_22 17 * 149 43 * 59 33_25 2543 50_7 50_1 46_21 13 * 197 11 * 233 17 * 151 7 * 367 31 * 83 47_10 2580 29 * 89 13 * 199 48_7 48_17 49 * 53 23 * 113 19 * 137 47_20 7 * 373 51_4 50_11 43 * 61 37 * 71 11 * 239 43_28 29 * 91 2640 19 * 139 43_14 11 * 241 7 * 379 49_16 50_3 2663 17 * 157 49_5 39_34 7 * 383 38_21 2687 40_33 47_22 51_2 2700 37 * 73 51_2 48_11 52_3 19 * 143 43_15 7 * 389 52_5 49_6 23 * 119 46_25 13 * 211 41 * 67 43_30 52_7 31 * 89 2760 11 * 251 42_17 17 * 163 47 * 59 44_29 7 * 397 23 * 121 50_17 31_30 51_14 49_20 34_27 7 * 401 53 * 53 29 * 97 49_11 2820 31 * 91 11 * 257 19 * 149 48_23 41_34 17 * 167 2843 37 * 77 45_14 51_16 50_19 7 * 409 47 * 61 19 * 151 17 * 169 48_23 2880 43 * 67 33_29 49 * 59 11 * 263 44_31 13 * 223 2903 53_10 41 * 71 54_1 23 * 127 37 * 79 2927 29 * 101 7 * 419 52_5 2940 17 * 173 7 * 421 13 * 227 53_12 46_29 11 * 269 2963 40_37 54_1 13 * 229 11 * 271 19 * 157 29 * 103 49 * 61 41 * 73 49_23
 + 1 + 7 + 11 + 13 + 17 + 19 + 23 + 29 +31 +37 + 41 + 43 + 47 + 49 + 53 + 59 3000 51_20 31 * 97 51_10 23 * 131 7 * 431 38_25 3023 13 * 233 7 * 433 54_11 55_4 17 * 179 11 * 277 45_32 43 * 71 23 * 133 3060 55_6 41_22 37 * 83 7 * 439 17 * 181 45_17 3083 55_8 11 * 281 19 * 163 7 * 443 29 * 107 13 * 239 47_30 11 * 283 51_14 3120 40_39 53 * 59 31 * 101 13 * 241 56_1 43 * 73 7 * 449 47 * 67 23 * 137 41 * 77 29 * 109 53_6 3167 55_12 19 * 167 17 * 187 3180 45_34 46_17 56_1 31 * 103 23 * 139 7 * 457 3203 53_20 19 * 169 56_9 55_14 11 * 293 7 * 461 50_27 53 * 61 41 * 79 3240 7 * 463 17 * 191 51_25 57_2 56_11 42_23 13 * 251 7 * 467 35_31 29 * 113 17 * 193 49 * 67 19 * 173 23 * 143 37 * 89 52_17 3300 49_30 57_1 43 * 77 57_8 31 * 107 50_13 3323 52_25 41_25 47 * 71 13 * 257 54_7 3347 17 * 197 7 * 479 57_2 3360 56_15 37 * 91 52_23 58_3 11 * 307 31 * 109 17 * 199 58_5 55_6 43 * 79 19 * 179 41 * 83 3407 7 * 487 58_7 13 * 263 3420 11 * 311 23 * 149 47 * 73 52_27 7 * 491 19 * 181 11 * 313 43_40 29 * 119 44_39 50_31 47_19 3467 45_38 23 * 151 49 * 71 3480 59 * 59 11 * 317 53_22 7 * 499 13 * 269 45_22 31 * 113 29 * 121 51_14 59_6 7 * 503 13 * 271 3527 48_35 58_13 59_1 3540 54_25 58_3 53 * 67 19 * 187 49_34 57_5 7 * 509 43 * 83 35_34 49 * 73 59_10 38_31 17 * 211 37 * 97 53_28 59 * 61 3600 13 * 277 49_18 23 * 157 43_42 44_41 47 * 77 3623 19 * 191 54_11 46_39 11 * 331 43_26 7 * 521 41 * 89 13 * 281 60_1 3660 7 * 523 19 * 193 55_17 48_37 59_14 13 * 283 29 * 127 31 * 119 39_31 49_36 55_26 23 * 161 11 * 337 53_30 47 * 79 57_10 3720 61 * 61 47_22 41 * 91 57_22 37 * 101 43_27 19 * 197 23 * 163 31 * 121 17 * 221 56_25 53 * 71 3767 60_13 49 * 77 55_26 3780 19 * 199 7 * 541 17 * 223 52_33 46_41 29 * 131 3803 13 * 293 37 * 103 11 * 347 61_10 42_29 43 * 89 7 * 547 53_32 11 * 349 3840 23 * 167 61_2 57_14 62_3 29 * 133 17 * 227 3863 53 * 73 49 * 79 54_31 59_20 11 * 353 23 * 169 60_17 17 * 229 7 * 557 3900 47 * 83 62_1 56_25 43 * 91 61_14 57_10 3923 52_35 51_19 31 * 127 7 * 563 59_7 3947 11 * 359 59 * 67 37 * 107 3960 17 * 233 58_9 19 * 209 29 * 137 41 * 97 23 * 173 7 * 569 58_25 13 * 307 7 * 571 49_40 39_34 4007 19 * 211 62_13 57_22
 + 1 + 7 + 11 + 13 + 17 + 19 + 23 + 29 +31 +37 + 41 + 43 + 47 + 49 + 53 + 59 4020 50_39 42_31 29 * 139 37 * 109 11 * 367 7 * 577 13 * 311 55_32 61_5 59_24 31 * 131 17 * 239 49 * 83 13 * 313 52_37 59_13 4080 53 * 77 61 * 67 63_2 58_27 17 * 241 63_2 11 * 373 7 * 587 39_35 23 * 179 13 * 317 31 * 133 4127 60_23 62_17 60_11 4140 41 * 101 29 * 143 7 * 593 48_43 59_26 57_13 23 * 181 11 * 379 43 * 97 64_9 37 * 113 47 * 89 53 * 79 59 * 71 7 * 599 19 * 221 4200 51_40 7 * 601 60_13 11 * 383 64_11 38_37 41 * 103 65_2 41_34 19 * 223 65_4 57_14 31 * 137 7 * 607 53_38 63_5 4260 65_6 17 * 251 61_11 57_32 47 * 91 11 * 389 4283 65_8 7 * 613 61_24 23 * 187 13 * 331 59 * 73 31 * 139 19 * 227 7 * 617 4320 29 * 149 62_7 61 * 71 7 * 619 49_44 55_18 43 * 101 50_43 19 * 229 66_1 49 * 89 53_21 11 * 397 17 * 257 62_23 29 * 151 4380 13 * 337 41 * 107 64_5 23 * 191 61_26 53 * 83 37 * 119 53_40 11 * 401 7 * 631 65_14 66_1 19 * 233 43 * 103 31 * 143 23 * 193 4440 60_29 39_38 60_23 61 * 73 64_19 49 * 91 4463 41 * 109 17 * 263 37 * 121 65_16 62_9 7 * 641 67 * 67 67_2 11 * 409 4500 7 * 643 57_17 13 * 347 48_47 49_46 47_30 4523 7 * 647 23 * 197 13 * 349 19 * 239 59 * 77 4547 65_18 29 * 157 47 * 97 4560 60_31 41_37 7 * 653 17 * 269 23 * 199 19 * 241 4583 13 * 353 59_15 54_41 43 * 107 51_26 17 * 271 11 * 419 7 * 659 31 * 149 4620 61_30 7 * 661 11 * 421 41 * 113 59_34 58_17 4643 68_5 65_6 56_39 59 * 79 49_29 13 * 359 29 * 161 68_7 64_11 4680 31 * 151 43 * 109 68_1 19 * 247 61 * 77 37 * 127 4703 17 * 277 7 * 673 53 * 89 64_25 46_33 29 * 163 52_45 58_37 7 * 677 4740 11 * 431 47 * 101 63_17 49 * 97 67 * 71 53_25 11 * 433 19 * 251 13 * 367 17 * 281 7 * 683 63_11 4787 55_42 68_13 67_5 4800 65_24 23 * 209 17 * 283 67_18 56_41 61 * 79 53 * 91 11 * 439 69_1 7 * 691 47 * 103 29 * 167 37 * 131 13 * 373 23 * 211 43 * 113 4860 69_10 31 * 157 63_22 11 * 443 61_34 41 * 119 19 * 257 67_20 67 * 73 59 * 83 29 * 169 69_2 7 * 701 70_3 17 * 289 63_25 4920 37 * 133 13 * 379 63_26 62_33 64_29 11 * 449 4943 49 * 101 46_35 69_14 41 * 121 7 * 709 4967 60_37 67_22 13 * 383 4980 17 * 293 57_22 31 * 161 63_32 19 * 263 62_15 5003 65_28 50_31 29 * 173 70_11 58_21 11 * 457 47 * 107 7 * 719 64_23
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Prepared by Matthias Engelhardt

last change: 2019-10-03