Find the number of polynomials P(x) with positive integer coefficients and with at most 2 terms such that P(2)=384.
Assume the polynomials with two terms are of the form: ax^e + bx^f and with e >= f
Initially allowing both terms to have equal exponents, I found 542: 536 with 2 terms, and 6 with 1 term. But ...
... I am reducing this number by 190 to:
346 with 2 terms, 6 with one term
a total of 352
The 190 have 2 terms but each term has the same exponent:
96 are of the form: (96+i)x + (96-i)x
48 are of the form: (48+i)x^2 + (48-i)x^2
24 are of the form: (24+i)x^3 + (48-i)x^3
12 are of the form: (12+i)x^4 + (48-i)x^4
6 are of the form: (6+i)x^5 + (48-i)x^5
3 are of the form: (3+i)x^6 + (48-i)x^6
1 is of the form: 2x^7 + 1x^7
This accounts for 190 where the exponents are the same. My opinion is that these are not in proper polynomial form and should not be counted. They are duplicative of the 6 polynomials with only one term.
The 6 polynomials with a single term are:
3x^7, 6x^6, 12x^5, 24x^4, 48x^3, 96x^2
The 346 polynomials with two terms (and different exponents, largest first) are:
1x^8 + 1x^7, 1x^8 + 2x^6, 1x^7 + 4x^6, 1x^8 + 4x^5, 1x^7 + 8x^5, 1x^8 + 8x^4, 1x^6 + 10x^5, 1x^7 + 16x^4, 1x^8 + 16x^3, 1x^6 + 20x^4, 1x^5 + 22x^4, 1x^7 + 32x^3, 1x^8 + 32x^2, 1x^6 + 40x^3, 1x^5 + 44x^3, 1x^4 + 46x^3, 1x^7 + 64x^2, 1x^8 + 64x^1, 1x^6 + 80x^2, 1x^5 + 88x^2, 1x^4 + 92x^2, 1x^3 + 94x^2, 1x^7 + 128x^1, 1x^6 + 160x^1, 1x^5 + 176x^1, 1x^4 + 184x^1, 1x^3 + 188x^1, 1x^2 + 190x^1, 2x^7 + 2x^6, 2x^7 + 4x^5, 2x^6 + 8x^5, 2x^7 + 8x^4, 2x^6 + 16x^4, 2x^7 + 16x^3, 2x^5 + 20x^4, 2x^6 + 32x^3, 2x^7 + 32x^2, 2x^5 + 40x^3, 2x^4 + 44x^3, 2x^6 + 64x^2, 2x^7 + 64x^1, 2x^5 + 80x^2, 2x^4 + 88x^2, 2x^3 + 92x^2, 2x^6 + 128x^1, 2x^5 + 160x^1, 2x^4 + 176x^1, 2x^3 + 184x^1, 2x^2 + 188x^1, 3x^6 + 6x^5, 3x^6 + 12x^4, 3x^5 + 18x^4, 3x^6 + 24x^3, 3x^5 + 36x^3, 3x^4 + 42x^3, 3x^6 + 48x^2, 3x^5 + 72x^2, 3x^4 + 84x^2, 3x^3 + 90x^2, 3x^6 + 96x^1, 3x^5 + 144x^1, 3x^4 + 168x^1, 3x^3 + 180x^1, 3x^2 + 186x^1, 4x^6 + 4x^5, 4x^6 + 8x^4, 4x^5 + 16x^4, 4x^6 + 16x^3, 4x^5 + 32x^3, 4x^6 + 32x^2, 4x^4 + 40x^3, 4x^5 + 64x^2, 4x^6 + 64x^1, 4x^4 + 80x^2, 4x^3 + 88x^2, 4x^5 + 128x^1, 4x^4 + 160x^1, 4x^3 + 176x^1, 4x^2 + 184x^1, 5x^6 + 2x^5, 5x^6 + 4x^4, 5x^6 + 8x^3, 5x^5 + 14x^4, 5x^6 + 16x^2, 5x^5 + 28x^3, 5x^6 + 32x^1, 5x^4 + 38x^3, 5x^5 + 56x^2, 5x^4 + 76x^2, 5x^3 + 86x^2, 5x^5 + 112x^1, 5x^4 + 152x^1, 5x^3 + 172x^1, 5x^2 + 182x^1, 6x^5 + 12x^4, 6x^5 + 24x^3, 6x^4 + 36x^3, 6x^5 + 48x^2, 6x^4 + 72x^2, 6x^3 + 84x^2, 6x^5 + 96x^1, 6x^4 + 144x^1, 6x^3 + 168x^1, 6x^2 + 180x^1, 7x^5 + 10x^4, 7x^5 + 20x^3, 7x^4 + 34x^3, 7x^5 + 40x^2, 7x^4 + 68x^2, 7x^5 + 80x^1, 7x^3 + 82x^2, 7x^4 + 136x^1, 7x^3 + 164x^1, 7x^2 + 178x^1, 8x^5 + 8x^4, 8x^5 + 16x^3, 8x^4 + 32x^3, 8x^5 + 32x^2, 8x^4 + 64x^2, 8x^5 + 64x^1, 8x^3 + 80x^2, 8x^4 + 128x^1, 8x^3 + 160x^1, 8x^2 + 176x^1, 9x^5 + 6x^4, 9x^5 + 12x^3, 9x^5 + 24x^2, 9x^4 + 30x^3, 9x^5 + 48x^1, 9x^4 + 60x^2, 9x^3 + 78x^2, 9x^4 + 120x^1, 9x^3 + 156x^1, 9x^2 + 174x^1, 10x^5 + 4x^4, 10x^5 + 8x^3, 10x^5 + 16x^2, 10x^4 + 28x^3, 10x^5 + 32x^1, 10x^4 + 56x^2, 10x^3 + 76x^2, 10x^4 + 112x^1, 10x^3 + 152x^1, 10x^2 + 172x^1, 11x^5 + 2x^4, 11x^5 + 4x^3, 11x^5 + 8x^2, 11x^5 + 16x^1, 11x^4 + 26x^3, 11x^4 + 52x^2, 11x^3 + 74x^2, 11x^4 + 104x^1, 11x^3 + 148x^1, 11x^2 + 170x^1, 12x^4 + 24x^3, 12x^4 + 48x^2, 12x^3 + 72x^2, 12x^4 + 96x^1, 12x^3 + 144x^1, 12x^2 + 168x^1, 13x^4 + 22x^3, 13x^4 + 44x^2, 13x^3 + 70x^2, 13x^4 + 88x^1, 13x^3 + 140x^1, 13x^2 + 166x^1, 14x^4 + 20x^3, 14x^4 + 40x^2, 14x^3 + 68x^2, 14x^4 + 80x^1, 14x^3 + 136x^1, 14x^2 + 164x^1, 15x^4 + 18x^3, 15x^4 + 36x^2, 15x^3 + 66x^2, 15x^4 + 72x^1, 15x^3 + 132x^1, 15x^2 + 162x^1, 16x^4 + 16x^3, 16x^4 + 32x^2, 16x^3 + 64x^2, 16x^4 + 64x^1, 16x^3 + 128x^1, 16x^2 + 160x^1, 17x^4 + 14x^3, 17x^4 + 28x^2, 17x^4 + 56x^1, 17x^3 + 62x^2, 17x^3 + 124x^1, 17x^2 + 158x^1, 18x^4 + 12x^3, 18x^4 + 24x^2, 18x^4 + 48x^1, 18x^3 + 60x^2, 18x^3 + 120x^1, 18x^2 + 156x^1, 19x^4 + 10x^3, 19x^4 + 20x^2, 19x^4 + 40x^1, 19x^3 + 58x^2, 19x^3 + 116x^1, 19x^2 + 154x^1, 20x^4 + 8x^3, 20x^4 + 16x^2, 20x^4 + 32x^1, 20x^3 + 56x^2, 20x^3 + 112x^1, 20x^2 + 152x^1, 21x^4 + 6x^3, 21x^4 + 12x^2, 21x^4 + 24x^1, 21x^3 + 54x^2, 21x^3 + 108x^1, 21x^2 + 150x^1, 22x^4 + 4x^3, 22x^4 + 8x^2, 22x^4 + 16x^1, 22x^3 + 52x^2, 22x^3 + 104x^1, 22x^2 + 148x^1, 23x^4 + 2x^3, 23x^4 + 4x^2, 23x^4 + 8x^1, 23x^3 + 50x^2, 23x^3 + 100x^1, 23x^2 + 146x^1, 24x^3 + 48x^2, 24x^3 + 96x^1, 24x^2 + 144x^1, 25x^3 + 46x^2, 25x^3 + 92x^1, 25x^2 + 142x^1, 26x^3 + 44x^2, 26x^3 + 88x^1, 26x^2 + 140x^1, 27x^3 + 42x^2, 27x^3 + 84x^1, 27x^2 + 138x^1, 28x^3 + 40x^2, 28x^3 + 80x^1, 28x^2 + 136x^1, 29x^3 + 38x^2, 29x^3 + 76x^1, 29x^2 + 134x^1, 30x^3 + 36x^2, 30x^3 + 72x^1, 30x^2 + 132x^1, 31x^3 + 34x^2, 31x^3 + 68x^1, 31x^2 + 130x^1, 32x^3 + 32x^2, 32x^3 + 64x^1, 32x^2 + 128x^1, 33x^3 + 30x^2, 33x^3 + 60x^1, 33x^2 + 126x^1, 34x^3 + 28x^2, 34x^3 + 56x^1, 34x^2 + 124x^1, 35x^3 + 26x^2, 35x^3 + 52x^1, 35x^2 + 122x^1, 36x^3 + 24x^2, 36x^3 + 48x^1, 36x^2 + 120x^1, 37x^3 + 22x^2, 37x^3 + 44x^1, 37x^2 + 118x^1, 38x^3 + 20x^2, 38x^3 + 40x^1, 38x^2 + 116x^1, 39x^3 + 18x^2, 39x^3 + 36x^1, 39x^2 + 114x^1, 40x^3 + 16x^2, 40x^3 + 32x^1, 40x^2 + 112x^1, 41x^3 + 14x^2, 41x^3 + 28x^1, 41x^2 + 110x^1, 42x^3 + 12x^2, 42x^3 + 24x^1, 42x^2 + 108x^1, 43x^3 + 10x^2, 43x^3 + 20x^1, 43x^2 + 106x^1, 44x^3 + 8x^2, 44x^3 + 16x^1, 44x^2 + 104x^1, 45x^3 + 6x^2, 45x^3 + 12x^1, 45x^2 + 102x^1, 46x^3 + 4x^2, 46x^3 + 8x^1, 46x^2 + 100x^1, 47x^3 + 2x^2, 47x^3 + 4x^1, 47x^2 + 98x^1, 48x^2 + 96x^1, 49x^2 + 94x^1, 50x^2 + 92x^1, 51x^2 + 90x^1, 52x^2 + 88x^1, 53x^2 + 86x^1, 54x^2 + 84x^1, 55x^2 + 82x^1, 56x^2 + 80x^1, 57x^2 + 78x^1, 58x^2 + 76x^1, 59x^2 + 74x^1, 60x^2 + 72x^1, 61x^2 + 70x^1, 62x^2 + 68x^1, 63x^2 + 66x^1, 64x^2 + 64x^1, 65x^2 + 62x^1, 66x^2 + 60x^1, 67x^2 + 58x^1, 68x^2 + 56x^1, 69x^2 + 54x^1, 70x^2 + 52x^1, 71x^2 + 50x^1, 72x^2 + 48x^1, 73x^2 + 46x^1, 74x^2 + 44x^1, 75x^2 + 42x^1, 76x^2 + 40x^1, 77x^2 + 38x^1, 78x^2 + 36x^1, 79x^2 + 34x^1, 80x^2 + 32x^1, 81x^2 + 30x^1, 82x^2 + 28x^1, 83x^2 + 26x^1, 84x^2 + 24x^1, 85x^2 + 22x^1, 86x^2 + 20x^1, 87x^2 + 18x^1, 88x^2 + 16x^1, 89x^2 + 14x^1, 90x^2 + 12x^1, 91x^2 + 10x^1, 92x^2 + 8x^1, 93x^2 + 6x^1, 94x^2 + 4x^1, 95x^2 + 2x^1
------------------------
count = 0
sols = []
for a in range(1,193):
for b in range(1,193):
for e in range(1,9):
# for e in range(6,7):
if a*2**e > 384:
continue
for f in range(1,e+1):
if e==f:
continue
# for f in range(6,7):
if b*2**f > 384:
continue
if e==f and a<b:
continue
if a*2**e + b*2**f == 384:
# print(a,b,e,f)
if e==f and e>6:
print(a,b,e,f,'asdf')
count += 1
text = (str(a) + 'x^' +
str(e) + ' + ' +
str(b) + 'x^' +
str(f))
sols.append(text)
print(count)
count2 = 0
sols2 = []
for a in range(1,192):
for e in range(1,9):
if a*2**e == 384:
count2 += 1
text = (str(a) + 'x^' +
str(e) )
sols2.append(text)
print(sols2)
print(count2)
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Posted by Larry
on 2025-02-17 14:37:15 |
Computer solution
